One-in-a-million

Suppose you can play on either of 2 slot machines:

  1. Slot machine A pays out with probability one in a million.
  2. Slot machine B pays out with probability one in 10.

Are you more likely to get a payout with one million attempts with slot machine A or with 10 attempts on slot machine B?

Have a think about this before scrolling down.

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I was prompted to think about this question by the following tweet, which includes both the answer and the relevant calculations.

So, there’s a bigger probability (0.65) that you’ll get a payout from 10 spins of slot machine B than from a million spins of slot machine A (probability 0.63).

Hopefully, the calculations above are self-explanatory. But just in case, here’s the detail. Suppose you have N attempts to win with a slot machine that pays out with probability 1/N.

  1. First we’ll calculate the probability of zero payouts in the N spins.

2. This means we get a zero payout on every spin.

3. The probability of a zero payout on one spin is one minus the probability of a win: 1 – 1/N.

4. So the probability of no payout on all the spins is

  (1-1/N)^N

  5. And the probability of at least one payout is

1- (1-1/N)^N

As explained in the tweet, with N=10 this gives 0.65 and with N=1,000,000 it gives 0.63. The tweet’s author explains in a follow-up tweet that he was expecting the same answer both ways.

But as someone in the discussion pointed out, that logic can’t be right. Suppose you had one attempt with slot machine C which paid out with probability 1. In other words, N=1 in my example above. Then, of course, you’d be bound to get a payout, so the probability of at least one payout is 1. So, although it’s initially perhaps surprising that you’re more likely to get a payout with 10 shots at slot machine B than with a million shots at slot machine A, the dependence on N becomes obvious when you look at the extreme case of slot machine C.


Footnote: What does stay the same in each case however is the average number of times you will win. With N shots at a slot machine with win probability 1/N, you will win on average once for any choice of N. Sometimes you’ll win more often, and sometimes you may not win at all (except when N=1). But the average number of wins if you play many times will always be 1.

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