Wednesday, 28 September 2016

open the box

I briefly mentioned the Monty Hall problem in an earlier discussion of probability, but because so many people cannot accept the problem’s conclusion, I thought that it might be worth taking a closer look.

In fact, the Monty Hall problem is a variant of Bertrand’s box paradox, which was first proposed by Joseph Bertrand in his 1889 book Calcul des probabilités. Imagine that you have three boxes, one of which contains two gold coins and one of which holds two silver coins. The third box contains one of each type of coin. You reach into one of the boxes, selected at random, and without looking inside, take out a gold coin. What is the probability that the remaining coin in the box you selected is also gold? The answer is not 1/2.

This is my analysis: the gold coin that is withdrawn is one of three possible cold coins. One of these possible gold coins will be from the box where the other coin is silver, while with the other two possibilities, the second coin in the box is also gold. In other words, the probability that the second, unseen, coin is also gold is 2/3.

This fact makes for an interesting, if dishonest, game, especially if played against people who have a poor understanding of probability. The ideal environment is a bar or pub. Take three beermats, which must be identically printed on both sides. Use a marker pen to draw an X on each side of the first beermat and an O on each side of the second. The third beermat should be marked with an X on one side and an O on the other.

Now shuffle the three beermats behind your back or under the table. Then place the three beermats on top of each other on the table. If you plan to play this game for money, it is probably a good idea to get someone else to do the shuffling. At this point, you announce that you can predict whether the mark on the other side of the top beermat is an X or an O.

Of course, you cannot do this every time, but if you say that the hidden mark is the same as the mark you can see, you have a 2/3 chance of being correct. In total, there are six marks, three X’s and three O’s. If the visible mark is an X, it will be one of three X’s. But two of those X’s are on a beermat where the other mark is also an X, and only one is on a beermat where the other mark is an O. Hence the probability of being correct of 2/3.

If you do plan to play this game for money, you may like to know that if you play a ‘best of three’ series, you are twice as likely to get it right twice as you are to get it wrong twice, and you are eight times more likely to make three correct guesses as you are to be wrong three times. In fact, the longer you play, the more the odds that you will come out ahead increase in your favour. Even with a best of three, those odds are already 20/27, which is more than 74 percent, although with a longer sequence you’re likely to be accused of cheating, especially if your victim thinks that it’s a straight 50/50 choice.

I will now return to the Monty Hall problem, which is named after a US gameshow host of the 1950s. Imagine that you are a contestant on a gameshow. There are three boxes, one of which contains £10,000. The other two are empty. You are asked to choose one of the boxes, and if you select the box with the money, the money’s yours. However, before you are allowed to open your chosen box, the host, who knows which box contains the money, steps forward and opens one of the other boxes to show that it is empty. Now comes the question: do you want to stick with your original choice? Or would you prefer to switch your choice to the other unopened box? You’d be foolish not to. There is a probability of two-thirds that you will be wrong with your first choice, and when you are wrong, the host has only one box that he can open to reveal its emptiness because the other one contains the money, as he well knows.

This marks the end of the serious stuff, but I did write a comic fantasy novel about 15 years ago that features an eccentric version of the Monty Hall problem. If you plan to read on, I should warn you that it is extremely silly.
As the evening drifted aimlessly onwards, the levels of hilarity, jollity and merriment rose and rose. There was only one possible cure. A late friend of the fat one used to say, disapprovingly, whenever the level of seriousness in a conversation looked likely to become dangerously high, that there are three subjects you should never discuss at parties: science, politics and religion, although it is difficult to see what the problem is. After all, scientists know the truth, priests tell the truth, and politicians hide the truth. Everyone knows that, but only Qumfl’quelunx himself could contrive to fashion all three into a perplexing, party-pooping pronouncement.

“I have a poseur!” he announced loudly, and then repeated himself, even louder, just in case anyone hadn’t heard him the first time.

No wonder his friend was always late. Word must have passed around. There was a stunned silence, in various combinations of shock and disbelief, topped off with a liberal sprinkling of horror. But Qumfl’quelunx had reached a bizarre conclusion. He had decided that there must be some question that the Grandmaster couldn’t answer. However, it couldn’t be something that he, Qumfl’quelunx, already knew the answer to, because then he’d know something that the Grandmaster didn’t know, which didn’t seem likely, so it would have to be something that he didn’t already know. And for that he interrupted his own party?

Anyway, the fat one has clearly been taking lessons in show business presentation (and bare-faced impudence) in his spare time from Dweebl’gulja, because this is what he announced to the less than expectant throng.

“A scientist offers you a wallet that, he says, he has proved contains one thousand pounds,” he began. “A priest offers you a wallet that, he says, contains one thousand pounds, but only if you have enough faith to believe that it does. A politician promises that he will show you a wallet at some unspecified date in the future, but he never mentions specific sums of money.”

“Now, here’s the difficulty,” continued His Plumpness majestically. “Only one wallet actually contains one thousand pounds. The other two are empty, which means that only one of the three actors is actually telling the truth. The problem is to choose the right wallet.”

“Excellent!” exclaimed Dweebl’gulja, although having made major inroads into his fifth glass of firewater could well have had some hand in the making of this serious error of judgement. “Party games? Now this is what I call a party!”

At any rate, this was the sort of perplexing poser that the Grandmaster usually dreams up, and it did not seem to be too difficult. But before Dweebl’gulja could offer his valuable opinion on this monetary mystery, Shunshelstinx insisted on pointing out that the politician’s promise was empty, as you would expect, because he had seen it being manipulated from behind a curtain.

Sneedl’bodja was even more peremptory, because he could follow the logic of the situation and the truth of the logic. He could? Must have been the firewater. He selected the scientist’s wallet.

It took Qumfl’quelunx to demonstrate the necessary leap of faith by selecting the priest’s wallet, but then he realized that he had posed the question and was thus not required to provide an answer. Never give an answer to a question that you’re never asked, he said to himself, in the process misremembering one of Dweebl’gulja’s favourite saws. So he waited for his guest to make a suggestion, which was the best the Grandmaster could do. Guess, in other words. And guess what? He also chose the priest’s wallet. If in doubt, he may have suggested, always take the priest’s wallet.

It was now time for His Plumpness to challenge the perspicacity of this pre-eminent pranking master, to assess the appropriateness of his analyses, to evaluate the exactitude of his estimates, to induce the incisiveness of his intuitive insights, to measure the magnitude of his mental machinations, to rule on the rectitude of his reasoning, to triangulate the thoroughness of his thinking. Or bemuse, baffle and bamboozle him with bullshit, more like.

“Now!” he announced with a flamboyant flourish, anxious to move on quickly before he could be overwhelmed by another avalanche of alliterative allusions. “Before you open your chosen wallet, I happen to know which wallet contains the money and which two are empty, so I’ll just open one of the empty wallets.”

With these words, he plucked a virtual wallet from the air, the one belonging to the scientist, and demonstrated that it contained nothing. No money. Not even enough for the bus fare home. And not even an explanation for why it had been wasting everyone’s time. But, then, that’s science for you.

“Now!” said Qumfl’quelunx, beginning to like the sound of sounding important. “Do you want to keep the priest’s wallet? Or would you rather believe the politician’s promise?” Now, presumably, you will know that in ordinary circumstances (no lying, in other words) the correct strategy if you want to maximize your chances of pocketing the money is to believe the politician’s promise and switch your allegiance. It is clearly to your advantage to change. If you cling on to the priest’s call to have faith, and to his wallet, your chance of being correct remains at a paltry one-third, not the half that you may have been led to believe by Shunshelstinx is the new probability. However, the probability is that you will have spotted his weakness in matters mathematical by now.

“Is it a trick question?” he asked for the twenty-seventh time.

But by exposing the obvious limitations of science, Qumfl’quelunx had somehow allowed the probability of a politician keeping a promise to surge to two-thirds, which isn’t very likely, as Dweebl’gulja was quick to point out. Unfortunately, the only explanation being offered is that the politician changed the rules halfway through, claiming despite damning evidence that no promises had been made in the first place, and that if promises had been made, it was all the fault of the previous party, which, we have been assured, did finish before midnight. But as a result of this perfidy, the politician’s party never recovered.

And neither did Qumfl’quelunx’s, although it does leave one small mystery unexplained. Just how did His Plumpness manage to evict his erstwhile ersatz erudition and replace it with the sort of cerebral conundrum that you would be more likely to associate with Dweebl’gulja? To avoid having to incorporate another coincidence into the already interplanetary preposterousness of the story, the management has decided that this curious oddity will have to remain unexplained, although if you must know, it has been rumoured, on very good authority, that Qumfl’quelunx has been up to his old tricks. Well, you wouldn’t expect him to be trying new tricks, he being like an old dog in that regard. And if he really has been meddling with the story again, it is certainly up to his usual standards of unsubtlety and very much of a piece with his lurid fashion sense. In keeping with his over-the-top style, he wouldn’t have been satisfied with the label ‘not stupid’, which even he could have been expected to carry off if he really tried, even though the only thing he can carry off with anything approaching panache is food. No, he would have to choose to masquerade as an ‘intellectual’. But perhaps the really odd thing is that he thinks nobody notices.
If, in spite of my prior warning, you have reached this point, you may possibly also enjoy A Problem with Hats from the same novel. It incorporates a well-known logic puzzle.

2 comments:

  1. With regard to Bertrand's box, after some head scratching,the answer is perhaps best understood by
    Left hand coin selected out of 2- gold coin box. right hand coin must be gold
    Right hand coin selected out of 2- gold coin box.left hand coin must be gold
    Gold coin selected out of 1 gold, 1 silver box, other coin must be silver
    So 2/3 is the correct answer
    However we must remember that M Bertrand was French while most of us most of us Brits tend to think of such conundrums as this in terms of Bookies Odds and may thus be confused by the 2/3 solution.To exacerbate the confusion, British bookies for some reason do not use odds of 2/3 or 3/2, but use 4/6 or 5/e instead
    Anyway couched as bookies odds, the 66.6% probability solution would actually be 1/2
    I know this is correct cos a Gelgin told me...

    ReplyDelete
    Replies
    1. Keith, you appear to be suggesting that Bertrand invented the terminology used in probability theory (although, to be fair, a lot of the early work in this field was done by two Frenchmen, Fermat and Pascal). And you should know better than to believe anything a gelgin told you, especially if that gelgin was Qumfl’quelunx.

      Delete

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