Christmas time is outlaw time

Christmas has always been a season for families, and it was great to have a visit from our youngest son, Nicky, and our daughter-out-law, Nicola, together known as ‘the Nicks’. His parents-out-law joined us for Christmas Eve to Boxing Day, leading to much good cheer and amusement, for instance in the form of well-lubricated charades, a game that declines in quality but improves in enjoyment as the alcoholic content climbs.

My son also took the opportunity to introduce me to the logical conundrum known as the Monty Hall problem. It’s called after a game show host, who at the climax of the game, called on the contestant to select one of three doors, behind two of which there was a goat, while behind the third was a car. While I might prefer a goat to certain cars I’ve come to know though not necessarily to love, you’ll understand that the premise of the game was that the contestant would win whatever was behind the door chosen and the desired outcome was to find the car.

Monty Hall and the game that left a problem

The special feature of the game is that after the contestant had chosen a door, the host, who knew what was behind all of them, would leave it shut but open another that revealed a goat. He would then offer the contestant the option of sticking with the first choice or changing.

At the heart of the Monty Hall problem is the question whether it’s better to stick or switch, or whether it makes no difference.

It seemed clear to me that there was nothing to choose between the options. That the probability of choosing the car was initially one in three and, with the elimination of one losing option, it had improved to one in two. To my embarrassment, it took me the best part of two hours to understand why this is not the case.

In my defence, I was in good company, as shown by the reaction to the journalist and populariser of mathematical oddities, Marilyn vos Savant, when she published the explanation: thousands, including many specialists, wrote to point out that she was wrong. But she wasn’t.

Vos Savant argued that it made much more sense to switch than to stick. In fact, switching doubles the chance of finding the car from one in three to two in three. It took me ages to grasp that, offering Nicky manifold opportunities for gentle but nonetheless mocking derision.

It’s one of those particularly irritating truths in mathematics that is completely counter-intuitive but utterly obvious once you’ve grasped it.

Over to you. If it takes you less than two hours to work out why in the Monty Hall game it’s twice as good a tactic to switch than to stick, then you’ll have beaten me and I congratulate you.

Not that you’ll have had as much fun as I did whiling away the dying moment of Christmas Eve over a few glasses with a mocking son and a bunch of outlaws.