11. The gambler’s-fallacy fallacy

A fair coin is tossed five times. By sheer chance, it lands heads the first four times, whereupon a man bets that it will come up tails on the fifth toss.

“After a run of heads,” he says, “tails is more likely.”

A woman rebukes the man for forgetting that the coin has no memory. The coin keeps no record of what happened on previous tosses, so heads is just as likely as tails on the fifth toss.

The man, she observes, has committed the gambler’s fallacy.

The next day, the man sees the woman walking to work. She has a car, so he asks why she does not drive. The woman replies that, while driving is more convenient, it is also more dangerous. As a compromise, she drives and walks on alternate days.

“I consider it too risky to drive every day,” she explains.

The man is shocked to hear this. After all, if she drives on Monday and nothing bad happens, this doesn’t mean that something bad is more likely to happen on Tuesday. The car keeps no record of what happened on Monday, so the risk of an accident on Tuesday is the same as it was on Monday. If she is willing to drive on Monday, she should be willing to drive on Tuesday too; indeed she should be willing to drive every day.

The man says that it is the woman’s turn to be committing the gambler’s fallacy. The woman denies this, however, and insists that this case is different, but each is unable to convince the other of their point of view.

On the third day, the woman spots the man trying to jaywalk across a busy two-lane road. It looks risky, so the woman suggests that he walk further down to where the road has just one lane, which would be safer to cross.

The man replies that it makes no difference whether you jaywalk across one lane of traffic or two. If you are prepared to cross one lane, he explains, then you can cross two lanes in good conscience too. Simply cross one lane first (which you are prepared to do) and then cross the remaining one lane (which you should again be prepared to do).

After all, if you cross the first lane safely, it will not suddenly become more dangerous to cross the second one. The road keeps no record of your previous crossings, so the risk of crossing the second lane (having crossed the first) is identical to that of crossing the first one. Since you were prepared to take the first risk, you should be prepared to take the second one too.

The woman is stunned to hear this. There is a big difference, she says, between risking something once and risking it twice in a row. Being prepared to jaywalk across one lane of traffic does not automatically mean being prepared to jaywalk across two lanes in succession. Indeed, if the man’s reasoning were sound, you should then be prepared to jaywalk across a road with any number of lanes, which would be absurd.

The man shakes his head and accuses the woman of falling for the gambler’s fallacy again, just as with walking to work on alternate days instead of simply driving every day.

The woman retorts that, on the contrary, the only fallacy here is to see the gambler’s fallacy where there is none.

Who is right and why?