What Is The Probability Of Getting Doubles With 2 Dice ? A Simple Probability Error Almost Everyone Makes

The following is from Joseph Mazur’s new book, What’s Luck Got to Do with It?:

…there is an authentically verified story that sometime in the 1950s a wheel in Monte Carlo came up even twenty-eight times in straight succession. The odds of that happening are close to 268,435,456 to 1. Based on the number of coups per day at Monte Carlo, such an event is likely to happen only once in five hundred years.

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Mazur uses this story to backup an argument which holds that, at least until very recently, many roulette wheels were not at all fair.

Assuming the math is right (we’ll check it later), can you find the flaw in his argument? The following example will help.

The Probability of Rolling Doubles

Imagine you hand a pair of dice to someone who has never rolled dice in her life. She rolls them, and gets double fives in her first roll. Someone says, “Hey, beginner’s luck! What are the odds of that on her first roll?”

Well, what are they?

There are two answers I’d take here, one much better than the other.

The first one goes like this. The odds of rolling a five with one die are 1 in 6; the dice are independent so the odds of rolling another five are 1 in 6; therefore the odds of rolling double fives are

$$(1/6)*(1/6) = 1/36$$.

1 in 36.

By this logic, our new player just did something pretty unlikely on her first roll.

But wait a minute. Wouldn’t ANY pair of doubles been just as “impressive” on the first roll? What we really should be calculating are the odds of rolling doubles, not necessarily fives. What’s the probability of that?

Since there are six possible pairs of doubles, not just one, we can just multiply by six to get 1/6. Another easy way to compute it: The first die can be anything at all. What’s the probability the second die matches it? Simple: 1 in 6. (The fact that the dice are rolled simultaneously is of no consequence for the calculation.)

Not quite so remarkable, is it?

For some reason, a lot of people have trouble grasping that concept. The chances of rolling doubles with a single toss of a pair of dice is 1 in 6. People want to believe it’s 1 in 36, but that’s only if you specify which pair of doubles must be thrown.

Now let’s reexamine the roulette “anomaly”

This same mistake is what causes Joseph Mazur to incorrectly conclude that because a roulette wheel came up even 28 straight times in 1950, it was very likely an unfair wheel. Let’s see where he went wrong.

There are 37 slots on a European roulette wheel. 18 are even, 18 are odd, and one is the 0, which I’m assuming does not count as either even or odd here.

So, with a fair wheel, the chances of an even number coming up are 18/37. If spins are independent, we can multiply probabilities of single spins to get joint probabilities, so the probability of two straight evens is then (18/37)*(18/37). Continuing in this manner, we compute the chances of getting 28 consecutive even numbers to be $$(18/37)^{28}$$.

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Turns out, this gives us a number that is roughly twice as large (meaning an event twice as rare) as Mazur’s calculation would indicate. Why the difference?

Here’s where Mazur got it right: He’s conceding that a run of 28 consecutive odd numbers would be just as interesting (and is just as likely) as a run of evens. If 28 odds would have come up, that would have made it into his book too, because it would be just as extraordinary to the reader.

Thus, he doubles the probability we calculated, and reports that 28 evens in a row or 28 odds in a row should happen only once every 500 years. Fine.

But what about 28 reds in a row? Or 28 blacks?

Here’s the problem: He fails to account for several more events that would be just as interesting. Two obvious ones that come to mind are 28 reds in a row and 28 blacks in a row.

There are 18 blacks and 18 reds on the wheel (0 is green). So the probabilities are identical to the ones above, and we now have two more events that would have been remarkable enough to make us wonder if the wheel was biased.

So now, instead of two events (28 odds or 28 evens), we now have four such events. So it’s almost twice as likely that one would occur. Therefore, one of these events should happen about every 250 years, not 500. Slightly less remarkable.

What about other unlikely events?

What about a run of 28 numbers that exactly alternated the entire time, like even-odd-even-odd, or red-black-red-black? I think if one of these had occurred, Mazur would have been just as excited to include it in his book.

These events are just as unlikely as the others. We’ve now almost doubled our number of remarkable events that would make us point to a broken wheel as the culprit. Only now, there are so many of them, we’d expect that one should happen every 125 years.

Finally, consider that Mazur is looking back over many years when he points out this one seemingly extraordinary event that occurred. Had it happened anytime between 1900 and the present, I’m guessing Mazur would have considered that recent enough to include as evidence of his point that roulette wheels were biased not too long ago.

That’s a 110-year window. Is it so surprising, then, that something that should happen once every 125 years or so happened during that large window? Not really.

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Slightly unlikely perhaps, but nothing that would convince anyone that a wheel was unfair.

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