Big Deal Or No Big Deal – How God Doesn’t Play Dice With The Universe. He Plays Poker
Clever people are a necessary evil. Necessary if you appreciate things such as Science, Technology and Art, that is.There is a view that if evolution hadn’t come up with clever people, there would only be a few thousand human beings scratching out a subsistence living and that Planet Earth would be a much pleasanter place for all the other life on it. This isn’t a very popular view though, so I’m not going to delve too far into it here. Most of us feel that we couldn’t do without i-Pod’s and smart phones and computers, so it seems that the intelligentsia are here to stay.
The cleverest person in the world at the moment is generally regarded as being Marilyn vos Savant, the highest estimate of her IQ being 228. She is unlikely to be robbed of this title in her lifetime. Mensa are the only people who give a toss about IQ scores nowadays and that’s only because they want more yearly subscriptions to their club. The rest of the world has long since figured out that IQ scores bear very little relation to your success and station in life if you don’t have the social skills to back up the brains. Sadly, high intelligence and good social skills are often uncomfortable bedfellows.
This point is clearly illustrated by the case of William Siddis, the man estimated to have the highest IQ ever recorded, at around about 300. Being ultra-brainy didn’t work out too well for him. His star burned out early in life. Graduated cum laude at Harvard when he was 16 and ended his days as an adding machine repair man. He died of a brain haemorrhage when he was 46, possibly as a result of thinking too hard.
Clever people have a habit of over thinking things. Probably the most prime example of this is the above mentioned Marilyn vos Savant’s infamous answer to the “Monty Hall Problem”, which she gave in Parade Magazine in 1990.
If you are unfamiliar with the Monty Hall Problem, it’s actually deceptively simple. It’s based on the endgame of “Let’s Make A Deal”, a game show hosted by Monty Hall between 1963 and 1979. In this endgame, the contestant stands in front of three doors which are numbered 1 to 3. Behind one door is something cool: a car. Behind each of the other two there is something not so cool: a goat.
The contestant picks a door, say door number one. This door remains shut. Monty, the host, then opens one of the other doors, say door number three. Behind the door is a goat. Monty doesn’t open this door at random. He knows what’s behind all the doors and doesn’t want you to get the car. So, of the two remaining doors, one has the car behind it and the other has a goat behind it. Monty asks the contestant “Do you want to stick with your original choice of door, or do you want to switch?” The contestant wins what is behind whichever door he picks.
So the question Marilyn was asked by one of her readers in parade magazine was “Do you switch or do you stick?” You would probably expect Marilyn’s answer to be “It doesn’t matter if you switch or not. There are two doors, one has a goat behind it and the other has a car behind it. The odds are 50/50.”
This is not the answer Marilyn gave though. What she actually said went something like this : “ Your original choice of door had a 1 in 3 chance of being the car. However, because one of the doors has been opened to reveal a goat, the conditions have changed and switching to the other door now gives you a 2 in 3 chance of winning the car. You should always switch”.
The world of Mathematics went nuts at this statement. Marilyn was accused of everything from never having read a probability textbook to being responsible for the decline in American educational standards. A bit of an over-reaction really. This was only a reply to a reader’s letter in Parade Magazine. She wasn’t claiming to have solved Fermat’s Last Theorem on the back of a beermat.
Here is an explanation of the problem from a mathematician who thinks Marilyn was right:
Personally, I disagree with both Marilyn’s answer and this mathematician’s explanation of her answer. They all start with some fundamentally wrong assumptions. The first is the assumption that the contestant wants to win the car. If I were a contestant on “Let’s Make A Deal”, the car would be no good to me. I don’t drive. I do have a massive back garden that I have difficulty keeping up with though, so a goat would be a much more useful prize for me to win. Besides, you can’t milk a car.
The second assumption is, of course, that the game isn’t rigged. I’ve seen pictures of this Monty Hall character and he looked like a bit of a wrong ‘un to me. This “three doors” scenario is just the old “pea and the walnut shells” con in another guise and that’s the oldest con in the book. As a contestant, I’d be wondering if there were any walls separating what was behind the doors. Might the car not be on castors and wheeled behind whatever door I happened not to pick? How do they stop the goats from bleating and revealing their positions to me. Do they tape their mouths shut or drug them or something?
Non-driving contestants and dodgy game show hosts aside, there is another more obvious reason I don’t agree with Marilyn vos Savant. Probabilities cease to be relevant the moment you pick that door. Before you pick that door, you have a 1 in 3 chance of getting the car and a 2 in 3 chance of getting the goat. After you pick the door, you have what you have.
These doors are not quantum doors. The host of the game show is Monty Hall, not Erwin Schrodinger. There’s not a probability wave behind them that collapses and resolves itself into either a car or a goat once a door is opened. When you choose a door at the beginning of the game, the die is cast. Choose the door with the goat behind it and there is a 100% of getting the goat. Choose the door with the car behind it and there is a 100% chance of getting the car. At that point, the game stops being a game of dice and turns into a hand of poker. Is the universe bluffing or has it got better cards than you?
All over America, High School maths geeks have been encouraged to perform experiments to see if sticking or switching is the best strategy. The results have been decisive. Play the game 100 times and you do indeed win by switching two thirds of the time. This evidence has been taken by many people to prove that Marilyn vos Savant’s theory was correct. It proves no such thing. A contestant on “Let’s Make A Deal” doesn’t get 100 goes at the prize, they only get one. One chance to win or lose.
If I were a contestant, with my one shot at winning the car – I could always sell it to buy a goat and pocket the change- here’s what I’d do. I’d realise that there was a one in three chance of me picking the door with the car behind it and the switching strategy not working. So I’d study old tapes of previous episodes of the show. Does Monty have any tells. Is there a particular facial expression when the contestant has picked the car? If the contestant picks the door with a goat behind it, how does Monty react? That kind of thing. And if I couldn’t spot any tell, I’d just trust my gut.
In Britain, we have resurrected “Let’s Make A Deal” in the form of Noel Edmonds’ “Deal Or No Deal.” This involves 22 boxes, sealed by an independent adjudicator. Each box contains a different sum of money ranging upwards from one penny to £250,000.( Well, the boxes actually contain pieces of cardboard with the sums of money written on them. It is a British version after all.) The chosen contestant picks a box and then one by one eliminates the rest one by one. If the contestant manages to eliminate enough boxes and reject the banker’s offers of money for their chosen box, they can end up in the position where there are only two boxes left. Their box and another box. They can even be in a situation where one of the boxes contains a penny and the other one contains £250,000.
Just like on “Let’s Make A Deal”, the contestant is offered a chance to switch. Noel Edmonds doesn’t know what’s in the boxes so the odds in this case actually are 50:50. But as any gambler will tell you, odds of 2:1 aren’t much better than 50:50. And whenever the contestant sticks to their guns because they’ve had a dream about it or they’ve chosen the birthday of their sick niece or something, they have a disproportionate tendency to win. They walk out of the television studio with a big fat cheque. All without the help of mathematicians or the planet’s most intelligent woman. Now that’s clever.
© Copyright Michael Grimes 2014