Here’s a fun probability problem that Benoit.Jottreau@smartodds.co.uk showed me. If you’re clever at probability, you might be able to solve it exactly; otherwise it’s easy to simulate. But as with previous problems of this type, I think it’s more interesting to find out what you would guess the answer to be, without thinking about it too deeply.

So, suppose you’ve got 10 coins. They’re fair coins, in the sense that if you toss any of them, they’re equally likely to come up heads or tails. You toss all 10 coins. You then remove the ones that come up heads. The remaining ones – the ones that come up tails – you toss again in a second round. Again, you remove any that come up heads, and toss again the ones that come up tails in a third round. And so on. In each round, you remove the coins that come up heads, and toss again the coins that come up tails. You stop once all of the coins have been removed.

The question: on average, how many rounds of this game do you need before all of the coins have been removed?

There are different mathematical ways of approaching this problem, but I’m not really interested in those. I’m interested in how good we are, collectively, at using our instincts to guess the solution to a problem of this type. So, I’d really appreciate it if you’d send me your best guess.

Actually, let’s make it a little more interesting. Can you send me an answer to a second question as well?

Second question: same game as above, but starting with 100 coins. This time, on average, how many rounds do you need before all of the coins have been removed?

I’ll discuss the answers you (hopefully) send me, and the problems themselves in more detail, in a subsequent post.

Please don’t fill out the survey if you solved the problem either mathematically or by simulation, though if you’d like to send me your solutions in either of those cases, I’d be very happy to look at them and discuss them with you.

# Needles, noodles and 𝜋

A while back, on Pi Day, I sent a post celebrating the number 𝜋 and mentioned that though 𝜋 is best known for its properties in connection with the geometry of a circle, it actually crops up all over the place in mathematics, including Statistics.

Here’s one famous example…

Consider a table covered with parallel lines like in the following figure.

For argument’s sake, let’s suppose the lines are 10 cm apart. Then take a bunch of needles – or matches, or something similar – that are 5 cm in length, drop them randomly onto the table, and count how many intersect one of the lines on the table.  Let’s suppose there are N needles and m of them intersect one of the lines. It turns out that N/m will be approximately 𝜋, and that the approximation is likely to improve if we repeat the experiment with a bigger value of N.

What this means in practice is that we have a statistical way of calculating 𝜋. Just do the experiment described above, and as we get through more and more needles, so the calculation of N/m is likely to lead to a better and better approximation of 𝜋.

There are various apps and so on that replicate this experiment via computer simulation, including this one, which is pretty nice. The needles which intersect any of the lines are shown in red; the others remain blue. The ratio N/m is shown in real-time, and if you’re patient enough it should get closer to the true value of 𝜋, the longer you wait. The approximation is also shown geometrically – the ratio N/m is very close to the ratio of a circle’s circumference to its diameter.

One important point though: the longer you wait, the greater will be the tendency for the approximation N/m to improve. However,  because of random variation in individual samples, it’s not guaranteed to always improve. For a while, the approximation might get a little worse, before inevitably (but perhaps slowly) starting to improve again.

In actual fact, there’s no need for the needles in this experiment to be half the distance between the lines. Suppose the ratio between the line separation and the needle length is r, then 𝜋 is approximated by

$\hat{\pi} = \frac{2rN}{m}$

In the simpler version above, r=1/2, which leads to the above result

$\hat{\pi} = \frac{N}{m}$

Now, although Buffon’s needle provides a completely foolproof statistical method of calculating 𝜋, it’s a very slow procedure. You’re likely to need very many needles to calculate 𝜋 to any reasonable level of accuracy. (You’re likely to have noticed this if you looked at the app mentioned above). And this is true of many statistical simulation procedures: the natural randomness in experimental data means that very large samples may be needed to get accurate results. Moreover, every time you repeat the experiment, you’re likely to get a different answer, at least to some level of accuracy.

Anyway… Buffon’s needle takes its name from Georges-Louis Leclerc, Comte de Buffon, a French mathematician in the 18th century who first posed the question of what the probability would be for a needle thrown at random to intersect a line. And Buffon’s needle is a pretty well-known problem in probability and Statistics.

Less well-known, and even more remarkable, is Buffon’s noodle problem. Suppose the needles in Buffon’s needle problem are allowed to be curved. So rather than needles, they are noodles(!) We drop N noodles – of possibly different shapes, but still 5 cm in length – onto the table, and count the total number of times the noodles cross a line on the table. Because of the curvature of the noodles, it’s now possible that a single noodle crosses a line more than once, so m is now the total number of line crossings, where the contribution from any one noodle might be 2 or more. Remarkably, it turns out that despite the curvature of the noodles and despite the fact that individual noodles might have multiple line crossings, the ratio N/m still provides an approximation to 𝜋 in exactly the same way it did for the needles.

This result for Buffon’s noodle follows directly from that of Buffon’s needle. You might like to try to think about why that is so. If not, you can find an explanation here.

Finally, a while back, I sent a post about Mendelian genetics. In it I discussed how Mendel used a statistical analysis of pea experiments to develop his theory of genetic inheritance. I pointed out, though, that while the theory is undoubtedly correct, Mendel’s statistical results were almost certainly too good to be true. In other words, he’d fixed his results to get the experimental results which supported his theory. Well, there’s a similar story connected to Buffon’s needle.

In 1901, an Italian mathematician, Mario Lazzarini, carried out Buffon’s needle experiment with a ratio of r=5/6. This seems like a strangely arbitrary choice. But as explained in Wikipedia, it’s a choice which enables the approximation of 355/113, which is well-known to be an extremely accurate fractional approximation for 𝜋. What’s required to get this result is that in a multiple of 213 needle throws, the same multiple of 113 needles intersect a line. In other words, 113 intersections when throwing 213 needles. Or 226 when throwing 426. And so on.

So, one explanation for Lazzarini’s remarkably accurate result is that he simply kept repeating the experiment in multiples of 213 throws until he got the answer he wanted, and then stopped. Indeed, he reported a value of N=3408, which happens to be 16 times 213. And in those 3408 throws, he reportedly got 1808 line intersections, which happens to be 16 times 113.

An alternative explanation is that Lazzarini didn’t do the experiment at all, but pretended he did with the numbers chosen as above so as to force the result to be the value that he actually wanted it to be. I know that doesn’t seem like a very Italian kind of thing to do, but there is some circumstantial evidence that supports this possibility. First, as also explained in Wikipedia:

A statistical analysis of intermediate results he reported for fewer tosses leads to a very low probability of achieving such close agreement to the expected value all through the experiment.

Second, Lazzarini reportedly described a physical machine that he used to carry out the experimental needle throwing. However, a basic study of the design of this machine shows it to be impossible from an engineering point of view.

So, like Mendel, it’s rather likely that Lazzarini invented some data from a statistical experiment just to get the answer that he was hoping to achieve. And the moral of the story? If you’re going to make evidence up to ‘prove’ your answer, build a little bit of statistical error into the answer itself, otherwise you might find statisticians in 100 years’ time proving (beyond reasonable doubt) you cheated.

# Love Island

A while back Harry.Hill@smartodds.co.uk gave a talk to the (then) quant team about trading strategies. The general issue is well-known: traders have to decide when to place a bet. Generally speaking they can place a bet early, when the price – the amount you get if you win the bet – is likely to be reasonably attractive. But in that case the liquidity of the market – the amount of money you can bet against – is likely to be low. Or they can wait until there is greater liquidity, but then the price is likely to be less attractive. So, given the option of a certain bet size at a stated price, should they bet now or wait in the hope of being able to make a bigger bet, albeit at a probably poorer price?

In general this is a difficult problem to tackle, and to make any sort of progress some assumptions have to be made about the way both prices and liquidity are likely to change as kick-off approaches. And Harry was presenting some tentative ideas, and pointing out some relevant research, that might enable us to get a handle on some of these issues.

Anyway, one of the pieces of work Harry referred to is a paper by F. Thomas Bruss, which includes the following type of example. You play a game where you can throw a dice (say) 10 times. Your objective is to throw a 6, at which point you can nominate that as your score, or continue.  But, here’s the catch: you only win if you throw a 6 and it’s the  final 6 in the sequence of 10 throws.

So, suppose you throw a 6 on the 3rd roll; should you stop? How about the 7th roll? Or the 9th? You can maybe see the connection with the trading issue: both problems require us to choose whether to stop or continue, based on an evaluation of the risk of what will subsequently occur.

Fast-forward a few days after Harry’s talk and I was reading Alex Bellos’s column in the Guardian. Alex is a journalist who writes about both football and mathematics (and sometimes both at the same time). His bi-weekly contributions to the Guardian take the form of mathematically-based puzzles. These puzzles are quite varied, covering everything from logic to geometry to arithmetic and so on. And sometimes even Statistics. Anyway, the puzzle I was reading after Harry’s talk is here. If you have time, take a read. Otherwise, here’s a brief summary.

It’s a basic version of Love Island. You have to choose from 3 potential love partners, but you only see them individually and sequentially. You are shown the first potential partner, and can decide to keep them or not. If you keep them, everything stops there. Otherwise you are shown the second potential partner. Again, you have to stick or twist: you can keep them, or you reject and are shown the third possibility. And in that case you are obliged to stick with that option.

In summary: once you stick with someone, that’s the end of the game. But if you reject someone, you can’t go back to them later. The question is: what strategy should you adopt in order to maximise the chances of choosing the person that you would have picked if you had seen all 3 at the same time?

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As well as giving a clearer description of the problem, Alex’s article also contains a link to his discussion of the solution. But what’s interesting is that it’s another example of an optimal stopping problem: once we’ve seen a new potential partner, and also previous potential partners, we have to make a decision on whether to stop with what we currently have, or risk trying to get an improvement in the future, knowing that we could also end up with something/someone worse. And if we can solve the problem for love partners, we are one step towards solving the problem for traders as well.

The Love Island problem discussed by Alex is actually a special case of The Secretary Problem.  A company needs to hire a secretary and does so by individual interviews. Once they’ve conducted an interview they have to hire or reject that candidate, without the possibility of returning to him/her once rejected. What strategy should they adopt in order to try to get the best candidate? In the Love Island version, there are just 3 candidates; in the more general problem, there can be any number. With 3 choices, and a little bit of patience, you can probably find the solution yourself (or follow the links towards Alex’s discussion of the solution). But how about if you had 1000 possible love partners? (Disclaimer: you don’t).

Actually, there is a remarkably simple solution to this problem whatever the number of options to choose from: whether it’s 3, 1000, 10,000,000 or whatever. Let this number of candidates be N. Then reject all candidates up to the M’th for some value of M, but keep note of the best candidate, C say, from those M options. Then accept the first subsequent candidate who is better than C in subsequent interviews (or the last candidate if none happens to be better).

But how to choose M? Well, even more remarkably, it turns out that if N is reasonably large, the best choice for M is around N/e, where $e \approx 2.718$ is a number that crops up a lot in mathematics. For N=1000 candidates, this means rejecting the first 368 and then choosing the first that is better than the best of those. And one more remarkable thing about this result: the probability that the candidate selected this way is actually the best out of all the available candidates is 1/e, or approximately 37%, regardless of the value of N.

With N=3, the value of N is too small for this approximate calculation of M to be accurate, but if you calculated the solution to the problem – or looked at Alex’s – you’ll see that the solution is precisely of this form, with M=2 and a probability of 50% of picking the best candidate overall.

Anyway, what I really like about all this is the way things that are apparently unconnected – Love Island, choosing secretaries, trading strategies – are fundamentally linked once you formulate things in statistical terms. And even if the solution in one of the areas is too simple to be immediately transferable to another, it might at least provide useful direction.

# The bean machine

Take a look at the following video…

It shows the operation of a mechanical device that is variously known as a bean machine, a quincunx or a Galton board. When the machine is flipped, a large number of small balls or beans fall through a funnel at the top of the device. Below the funnel is a layered grid of pegs. As each bean hits a peg it can fall left or right – with equal probability if the board is carefully made – down to the next layer, where it hits another peg and can again go left or right. This repeats for a number of layers, and the beans are then collected in groups, according to the position they fall in the final layer. At the end you get a kind of physical histogram, where the height of the column of beans corresponds to the frequency with which the beans have fallen in that slot.

Remarkably, every time this experiment is repeated, the pattern of beans at the bottom is pretty much the same: it’s symmetric, high in the middle, low at the edges and has a kind of general bell-shape. In fact, the shape of this histogram will be a good approximation to the well-known normal distribution curve:

As you probably know, it turns out that the relative frequencies of many naturally occurring phenomena look exactly like this normal curve: heights of plants, people’s IQ, brightness of stars…. and indeed (with some slight imperfections) the differences in team points in sports like basketball.

Anyway, if you look at the bottom of the bean machine at the end of the video, you’ll see that the heights of the columns of beans – which in itself represents the frequency of beans falling in each position – resembles this same bell-shaped curve. And this will happen – with different small irregularities – every time the bean machine is re-started.

Obviously, just replaying the video will always lead to identical results, so you’ll have to take my word for it that the results are similar every time the machine is operated. There are some simulators available, but my feeling is you lose something by not seeing the actual physics of real-world beans falling into place. Take a look here if you’re interested, though I suggest you crank the size and speed buttons up to their maximum values first.

But why should it be that the bean machine, like many naturally occurring phenomena, leads to frequencies that closely match the normal curve?

Well, the final position of each bean is the result of several random steps in which the bean could go left or right. If we count +1 every time the bean goes right and -1 every time the bean goes left, then the final position is the sum of these random +/-1 outcomes. And it turns out, that under fairly general conditions, that whenever you have a process that is the sum of several random experiments, the final distribution is bound to look like this bell-shaped normal curve.

This is a remarkable phenomenon. The trajectory of any individual bean is unpredictable. It could go way to the left, or way to the right, though it’s more likely that it will stay fairly central. Anything is possible, though some outcomes are more likely than others. However, while the trajectory of individual beans is unpredictable, the collective behaviour of several thousand beans is entirely predictable to a very high degree of accuracy: the frequencies within any individual range will match very closely the values predicted by the normal distribution curve. This is really what makes statistics tick. We can predict very well how a population will behave, even if we can’t predict how individuals will behave.

Even more remarkably, if the bean machine has enough layers of pegs, the eventual physical histogram of beans will still look like the normal distribution curve, even if the machine has some sort of bias. For example, suppose the beans were released, but that the machine wasn’t quite vertical, so that the beans had a higher tendency to go left, rather than right, when they hit a peg. In this case, as long as there were sufficiently many layers of pegs, the final spread of beans would still resemble the normal curve, albeit no longer centred at the middle of the board. You can try this in the simulator by moving the left/right button away from 50%.

Technically, the bean machine is a physical illustration of a mathematical result generally termed the Central Limit Theorem. This states that in situations like those illustrated by the bean machine, where a phenomenon can be regarded as a sum of random experiments, then under general conditions the distribution of final results will look very much like the well-known bell-shaped normal curve.

It’s difficult to overstate the importance of this result – which is fundamental to almost all areas of statistical theory and practice – since it lets us handle probabilities in populations, even when we don’t know how individuals behave. And the beauty of the bean machine is that it demonstrates that the Central Limit Theorem is meaningful in the real physical world, and not just a mathematical artefact.

Can’t live without your own desktop bean machine? I have good news for you…

# Pulp Fiction (Our Esteemed Leader’s cut)

The previous post had a cinematic theme. That got me remembering an offsite a while back where Matthew.Benham@smartbapps.co.uk gave a talk that I think he called ‘Do the Right Thing’, which is the title of a 1989 Spike Lee film. Midway through his talk Matthew gave a premiere screening of his own version of a scene from Pulp Fiction. Unfortunately, I’ve been unable to get hold of a copy of Matthew’s cut, so we’ll just have to make do with the inferior original….

The theme of Matthew’s talk was the importance of always acting in relation to best knowledge, even if it contradicts previous actions taken when different information was available. So, given the knowledge and information you had at the start of a game, you might have bet on team A. But if the game evolves in such a way that a bet on team B becomes positive value, you should do that. Always do the right thing. And the point of the scene from Pulp Fiction? Don’t let pride get in the way of that principle.

These issues will make a great topic for this blog sometime. But this post is about something else…

Dependence is a big issue in Statistics, and we’re likely to return to it in different ways in future posts. Loosely speaking, two events are said to be independent if knowing the outcome of one, doesn’t affect the probabilities of the outcomes of the other. For example, it’s usually reasonable to treat the outcomes of two different football matches taking place on the same day as independent. If we know one match finished 3-0, that information is unlikely to affect any judgements we might have about the possible outcomes of a later match. Events that are not independent are said to be dependent: in this case, knowing the outcome of one will affect the outcome of the other.  In tennis matches, for example, the outcome of one set tends to affect the chances of who will win a subsequent set, so set winners are dependent events.

With this in mind, let’s follow-up the discussion in the previous 2 posts (here and here) about accumulator bets. By multiplying prices from separate bets together, bookmakers are assuming that the events are independent. But if there were dependence between the events, it’s possible that an accumulator offers a value bet, even if the individual bets are of negative value. This might be part of the reason why Mark Kermode has been successful in several accumulator bets over the years (or would have been if he’d taken his predictions to the bookmaker and actually placed an accumulator bet).

Let me illustrate this with some entirely made-up numbers. Let’s suppose ‘Pulp Fiction (Our Esteemed Leader’s cut)’, is up for a best movie award, and its upstart director, Matthew Benham, has also been nominated for best director. The numbers for single bets on PF and MB are given in the following table. We’ll suppose the bookmakers are accurate in their evaluation of the probabilities, and that they guarantee themselves an expected profit by offering prices that are below the fair prices (see the earlier post).

True Probability Fair Price Bookmaker Price
Best Movie: PF 0.4 2.5 2
Best Director: MB 0.25 4 3.5

Because the available prices are lower than the fair prices and the probabilities are correct, both individual bets have negative value (-0.2 and -0.125 respectively for a unit stake). The overall price for a PF/MB accumulator bet is 7, which assuming independence is an even poorer value bet, since the expected winnings from a unit stake are

$0.4 \times 0.25 \times 7 -1 = -0.3$

However, suppose voters for the awards tend to have similar preferences across categories, so that if they like a particular movie, there’s an increased chance they’ll also like the director of that movie. In that case, although the table above might be correct, the probability of MB winning the director award if PF (MB cut) is the movie winner is likely to be greater than 0.25. For argument’s sake, let’s suppose it’s 0.5. Then, the expected winnings from a unit stake accumulator bet become

$0.4 \times 0.5 \times 7 -1 = 0.4$

That’s to say, although the individual bets are still both negative value, the accumulator bet is extremely good value. This situation arises because of the implicit assumption of independence in the calculation of accumulator prices. When the assumption is wrong, the true expected winnings will be different from those implied by the bookmaker prices, potentially generating a positive value bet.

Obviously with most accumulator bets – like multiple football results – independence is more realistic, and this discussion is unhelpful. But for speciality bets like the Oscars, or perhaps some political bets where late swings in votes are likely to affect more that one region, there may be considerable value in accumulator bets if available.

If anyone has a copy of Our Esteemed Leader’s cut of the Pulp Fiction scene on a pen-drive somewhere, and would kindly pass it to me, I will happily update this post to include it.

# How to not win ￡194,375

In the previous post we looked at why bookmakers like punters to make accumulator bets: so long as a gambler is not smart enough to be able to make positive value bets, the bookmaker will make bigger expected profits from accumulator bets than from single bets. Moreover, even for smart bettors, if any of their individual bets are not smart, accumulator bets may also favour the bookmaker.

With all this in mind, here’s a true story…

Mark Kermode is a well-known film critic, who often appears on BBC TV and radio. In the early 90’s he had a regular slot on Danny Baker’s Radio 5 show, discussing recent movie releases etc. On one particular show early in 1992, chatting to Danny, he said he had a pretty good idea of how most of the important Oscars would be awarded that year. This was actually before the nominations had been made, so bookmaker prices on award winners would have been pretty good and since Radio 5 was a predominantly sports radio station, Danny suggested Mark make a bet on the basis of his predictions.

Fast-forward a few months to the day after the Oscar awards and Danny asked Mark how his predictions had worked out. Mark explained that he’d bet on five of the major Oscar awards and they’d all won. Danny asked Mark how much he’d won and he replied that he’d won around ￡120 for a ￡25 stake.  Considering the difficulty in predicting five correct winners, especially before nominations had been made, this didn’t seem like much of a return, and Danny Baker was incredulous. He’d naturally assumed that Mark would have placed an accumulator bet with the total stake of ￡25, whereas what Mark had actually done was place individual bets of ￡5 on each of the awards.

Now, I’ve no idea what the actual prices were, but since the bets were placed before the nominations were announced, it’s reasonable to assume that the prices were quite generous. For argument’s sake, let’s suppose the bets on each of the individual awards  had a price of 6. Mark then placed a ￡5 bet on each, so he’d have made a profit of ￡25 per bet, for an overall profit of ￡125. Now suppose, instead, he’d made a single accumulator bet on all 5 awards. In this case he’d have made a profit of

$\pounds 25 \times 6 \times 6 \times 6 \times 6 \times 6 -\pounds 25 = \pounds 194,375$

Again, I’ve no idea if these numbers are accurate or not, but you get the picture. Had Mark made the accumulator bet that Danny intended, he’d have made a pretty big profit. As it was, he won enough for a night out with a couple of mates at the cinema, albeit with popcorn included.

Of course, the risk you take with an accumulator is that it just takes one bet to fail and you lose everything. By placing 5 single bets Mark would still have won ￡95 if one of his predictions had been wrong, and would even make a fiver if he got just one prediction correct. But by not accumulating his bets, he also avoided the possibility of winning ￡194,375 if all 5 bets came in. Which they did!

So, what’s the story here? Though an accumulator is a poor value bet for mug gamblers, it may be an extremely valuable bet for sharp gamblers, and the evidence suggests (see below) that Mark Kermode is sharper than the bookmakers for Oscar predictions.

Is Mark Kermode really sharper than the bookmakers for Oscar predictions? Well, here’s a list  of his predictions for the main 6 (not 5) categories for the years 2006-2017. Mark predicted all 6 categories with 100% accuracy twice in twelve years. I guess that these predictions weren’t always made before the nominations, so the prices are unlikely to be as good as in the example described above. But still, the price on a 6-fold accumulator will have been pretty good regardless. And he’d have won twice, in addition to the 1992 episode (and possibly more often in the intervening years for which I don’t have data). Remarkably, he would have won again in 2017 if the award for best movie had gone to La La Land, as was originally declared winner, rather than Moonlight, which was the eventual winner.

Moral: try to find out Mark’s predictions for the 2019 Oscars and don’t make the mistake of betting singles!

And finally, here’s Mark telling the story of not winning something like￡194,375 in his own words:

# Bookmakers love accumulators

You probably know about accumulator, or so-called ‘acca’, bets. Rather than betting individually on several different matches, in an accumulator any winnings from a first bet are used as the stake in a second bet.  If either bet loses, you lose, but if both bets win, there’s the potential to make more money than is available from single bets due to the accumulation of the prices. This process can be applied multiple times, with the winnings from several bets carried over as the stake to a subsequent bet, and the total winnings if all bets come in can be substantial. On the downside, it just takes one bet to lose and you win nothing.

Bookmakers love accumulators, and often apply special offers – as you can see in the profile picture above – to encourage gamblers to make such bets. Let’s see why that’s the case.

Consider a tennis match between two equally-matched players. Since the players are equally-matched, it’s reasonable to assume that each has a probability 0.5 of winning. So if a bookmaker was offering fair odds on the winner of this match, he should offer a price of 2 on either player, meaning that if I place a bet of 1 unit I will receive 2 units (including the return of my stake) if I win. This makes the bet fair, in the sense that my expected winnings – the amount I would win on average if the game were repeated  many times – is zero. This is because

$(1/2 \times 2) + (1/2 \times 0) -1 = 0$

That’s the sum of the probabilities multiplied by the prices, take away the stake.

The bet is fair in the sense that, if the match were repeated many times, both the gambler and the bookmaker would expect neither to win nor lose. But bookmakers aren’t in the business of being fair; they’re out to make money and will set lower prices to ensure that they have long-run winnings. So instead of offering a price of 2 on either player, they might offer a price of 1.9. In this case, assuming gamblers split their stakes evenly across two players, bookmakers will expect to win the following proportion of the total stake

$1-1/2\times(1/2 \times 1.9) - 1/2\times (1/2 \times 1.9)=0.05$

In other words, bookmakers have a locked-in 5% expected profit. Of course, they might not get 5%. Suppose most of the money is placed on player A, who happens to win. Then, the bookmaker is likely to lose money. But this is unlikely: if the players are evenly matched, the money placed by different gamblers will probably be evenly spread between the two players. And if it’s not, then the bookmakers can adjust their prices to try to encourage more bets on the less-favoured side.

Now, in an accumulator bet, the prices are multiplied. It’s equivalent to taking all of your winnings from a first bet and placing them on a second bet. Then those winnings are placed on the outcome of a third bet, and so on. So if there are two tennis matches, A versus B and C versus D, each of which is evenly-matched, the fair and actual prices on the accumulator outcomes are as follows:

Accumulator Bet A-C A-D B-C B-D
Fair Price 4 4 4 4
Actual Price 3.61 3.61  3.61 3.61

The value 3.61 comes from taking the prices of the individual bets, 1.9 in each case, and multiplying them together. It follows that the expected profit for the bookmaker is

$1-4\times 1/4\times(1/4 \times 3.61) = 0.0975$.

So, the bookmaker profit is now expected to be almost 10%. In other words, with a single accumulator, bookmakers almost double their expected profits. With further accumulators, the profits increase further and further. With 3 bets it’s over 14%; with 4 bets it’s around 18.5%. Because of this considerable increase in expected profits with accumulator bets, bookmakers can be ‘generous’ in their offers, as the headline graphic to this post suggests. In actual fact, the offers they are making are peanuts compared to the additional profits they make through gamblers making accumulator bets.

However… all of this assumes that the bookmaker sets prices accurately. What happens if the gambler is more accurate in identifying the fair price for a bet than the bookmaker? Suppose, for example, a gambler reckons correctly that the probabilities for players A and C to win are 0.55 rather than 0.5. A single stake bet spread across the 2 matches would then generate an expected profit of

$0.55\times(1/2 \times 1.9) + 0.55\times (1/2 \times 1.9) -1 = 0.045$

On the other hand, the expected profit from an accumulator bet on A-C is

$(0.55\times1.9) \times (0.55\times1.9) -1 = 0.092$

In other words, just as the bookmaker increases his expected profit through accumulator bets when he has an advantage per single bet, so does the gambler. So, bookmakers do indeed love accumulators, but not against smart gamblers.

In the next post we’ll find out how not knowing the difference between accumulator and standard bets cost one famous gambler a small fortune.

Actually, the situation is not quite as favourable for smart gamblers as the above calculation suggests. Suppose that the true probabilities for a win for A and C are 0.7 and 0.4, which still averages at 0.55. This situation would arise, for example, if the gambler was using a model which performed better than he realised for some matches, but worse than he realised for others.

The expected winnings from single bets remain at 0.045. But now, the expected winnings from an accumulator bet are just:

$(0.7\times1.9) \times (0.4\times1.9) -1 = 0.011,$

which is considerably lower. Moreover, with different numbers, the expected winnings from the accumulator bet could be negative, even though the expected winnings from separate bets is positive. (This would happen, for example, if the win probabilities for A and C were 0.8 and 0.3 respectively.)

So unless the smart gambler is genuinely smart on every bet, an accumulator bet may no longer be in his favour.

# Stuck in jail?

In an earlier post, Get out of jail,  I set the following problem:

If I roll a standard dice until I get a 6, how many rolls of the dice will I need on average?

A summary of the 8 answers I received is given in the figure below:.

So, 3 people got the answer right, perhaps because they know the general theory which leads to this answer. All other respondents under-estimated the answer, perhaps not taking into account that the number of throws needed could be 10, 20 or, in theory, even more.

But maybe I wasn’t fair in the question, since ‘average’ can have different meanings. The usual interpretation is the ‘mean’, and it’s the mean which takes the value 6. But another choice is the median, which for this problem is 4: on roughly 50% of occasions you’d need 4 throws or less to get a 6, and on a similar amount of occasions you’d need 5 throws or more. So, if you interpreted my question as asking for the median, you were right if your answer was 4, and close if your answer was 3. So again, everyone did really well by one interpretation of the problem. (Of course, if your answer was 3 because you divided 6 by 2 you were close, but for the wrong reason).

The reason why the mean and the median are so different in this problem can be seen in the following figure, which shows the probability for every possible number of throws from 1 to 25. Beyond 25, the probabilities are all very close to zero.

If you imagine the lines showing the probabilities as being made of strips of metal lying flat on a piece of paper, then the mean is the line of equilibrium – i.e. if I place my finger vertically at the mean – so, 6 in this case – the figure would balance perfectly. (Or at least it would if I hadn’t truncated the graph at 25 – in reality the vertical lines stretch out to infinity, but with smaller and smaller heights).

In contrast, the median is the vertical line at which the total lengths of the bars are equally balanced on each side. We can’t get this perfectly, because of the jumps in the lengths from one bar to the next, but very roughly the sum of the lengths of the bars up to the fourth one is equal to the sum of the lengths from the fifth one onwards, so the median is 4.

Consequently, the mean and median are points of equilibrium by different physical rules – by weight for the mean and by total length for the median –  and when probability distributions are very asymmetric, as in the figure above, the values can be quite different.

Anyway, I’d intended to ask for the mean number of throws required, and the answer to that is 6. But why?

It’s not a proper proof, but suppose I rolled the dice 6 million times. Because of the symmetry of the dice, you’d expect around 1 million of those throws to come up 6. And those 1 million 6’s will be randomly spread among the 6 million throws. So, on average, the 6’s will be 6 throws apart. In other words: you have to wait an average of 6 throws after rolling a 6 to throw another 6. And by similar reasoning, you’d have to wait an average of 6 throws before getting the first 6.

Obviously, there’s nothing special about the number 6 here. Or about dice. In general, if I repeat an experiment where there are $N$ different possible outcomes, each of which is equally likely, the average number of times I’ll have to repeat the experiment before having a success is $N$. For example, if cereal packs contain a gift, and there are 10 different available gifts, I’ll need an average of 10 cereal packs to get any particular gift that I’m hoping for.

Just for completeness, and it’s entirely optional, here’s a formal mathematical proof.

Let the average number of throws required be $A$.

On the first throw of the dice there are 2 possibilities:

• I get a 6.
• I don’t get a 6.

These possibilities occur with probability 1/6 and 5/6 respectively. In the first case, I’ve achieved the objective of rolling a 6, and so the total number of throws needed was 1. In the second case, I haven’t achieved the objective, and so still have to make $A$ throws on average to get a 6, on top of the throw that I’ve already made. In other words, in this case I will have to make a total of $A+1$ throws on average. So, with probability 1/6 I just need 1 throw, but with probability 5/6 I need an average of $A+1$ throws. But we also know that the average number of throws overall is $A$. So

$A =1/6 \times 1 +(5/6)\times (A+1)$

This gives an equation to solve for $A$, and you can easily check that it works with $A=6$.

One more quick aside based on the responses to the original post: it’s obviously difficult to draw many conclusions from just 8 responses, though I’m grateful to those of you who did respond. Clearly this type of post isn’t generating much interest, and maybe that’s true of the blog as a whole. I’m planning to give the blog a bit of a break over Christmas anyway, but before then I’ll include a post inviting feedback so that I can try to push the blog in a different direction if that’s preferred. Or maybe just wind things up if that seems more appropriate.

# Borel

Struggling for ideas for Christmas presents? Stuck with an Amazon voucher from your employer and don’t know what to do with it? No idea how you’re going to get through Christmas with the in-laws? Trying to ‘Gamble Responsibly‘ but can’t quite kick the habit?

You can thank me later, but I have the perfect solution for you:

Borel

This is a new Trivial-Pursuit-style board game, but with a twist. Players are given a question involving dice, coloured balls or some other experimental apparatus, and have to bet on the outcome. There’s not enough time to actually do the probability calculations, so you just have to go with intuition. You can make bets of different sizes and, just like in real life, should make bigger bets when you think the odds are more in your favour.

This is part of the description at Amazon:

The game combines the human mind’s difficulty to deal with probabilistic dilemmas with the strategic thinking of competitive gambling.

And:

It is designed to reward probabilistic reasoning, intuition, strategic thinking and risk-taking!

In other words, it’s just like Smartodds-world, but without models to help you.

Disclaimer: The description and reviews look great, and I’ve ordered a set for myself, but I haven’t played it yet. I’ll try it on my family over Christmas and let you know how we get on. If you want a set for yourself or your loved ones, it’s available on Amazon here.

# Get out of jail

As a rule, I don’t intend to use this blog to cover standard theory in probability or Statistics. But for a number of reasons, which will become clear in future posts, I’d like you to think about the following problem.

In the game of Monopoly, when you’re in jail, one way of getting out is to throw equal numbers – a so-called double – on the two dice. Actually, in standard Monopoly rules, you’re only allowed to try this method 3 times, and are then required to pay your way out, unless you have a ‘Get out of Jail Free’ card. But suppose there weren’t any such limit to the number of throws you could take, and just kept taking further turns until you rolled a double. In that case, how many rolls, on average, would it take for you to roll a double and get out of jail?

It’s very easy to show that the probability of throwing a double on any one throw is 1/6: let the score on the first dice be S; then the two dice will have the same score if the second dice also shows S. And the probability of that, by symmetry of the dice, is 1/6.

So actually, since 1/6 is also the probability of throwing a 6 on a single dice, this problem has an identical solution to the following one:

If I roll a standard dice until I get a 6, how many rolls of the dice will I need on average?

This is a standard problem in probability and statistics, so anyone who’s studied statistics to a reasonable level will automatically know the answer. But if you don’t know the answer, use your intuition to guess what it might be. Either way, please send me your answer via this survey.

I’ll discuss the solution – and your guesses – in a future post. Like I say, I’ll also be making use of the result in a couple of different ways, also in future posts.