Take It or Leave It

I like gameshows a lot. I like looking at the maths behind decisions in games. I also like making up hypothetical situations and working out the best case scenario.

Take It or Leave It

In the UK, Take It or Leave It was broadcast on Challenge. It is not to be confused with the French version of Deal or No Deal called À prendre ou à laisser, which translates to Take it or Leave It. Anyway, a synopsis of the show can be found using the magic of Google.

The Assumption: Everyone playing knows the answers to all of the first ten questions that get asked.

The Scenario: You and your friend have succesfully answered the first 8 questions, and have banked a tasty £22,500. Only the £15,000 and £12,500 remain uncovered but also so do both booby traps. The question comes up "Who was the 13th undisputed FIDE world chess champion?" You are offered Bobby Fischer. Do you take it or leave it? Answer after the break...



The answer is of course Garry Kasparov, so you should leave it. But it's a surprisingly close decision! To get into the final you will either have the first safe picker pick money both times, leaving the booby traps to the second safe picker (that's the perfect £50,000 banked!), or the first safe picker picks a booby trap, the second safe picker picks cash, then the other pair lose via booby trap. (that's the still handy £22,500 in the jackpot).

Every other scenario is game over. The jackpot will have different amounts in, but as it's for the other pair we don't care.

As we are picking items into specific places, and are treating the two cash safes as equal (the two booby traps are equal anyway) we can use combinations. We have 4 safes, and are picking 2 of them to be the first safe picks on each of question 9 and 10. So 4 choose 2, that's 4! / 2! * (4-2)! giving 6 different ways, and all 6 ways have an equal chance of happening.

We have a 1/6 chance of getting to the final with £50,000 and 1/6 of getting to the final of £22,500. That gives us an expected value of £12,083.33.

But what if we deliberately give the wrong answer? The other pair has to give a correct answer on question 10, then they have a 2 in 4 chance of hitting a booby trap. The jackpot stays at £22,500 so as we get a chance to play for that half the time out expected value is £11,250.

So there we have it. Purely looking at the expected value, it's slightly better to give correct answers and hope. There is a slightly greater chance of getting to the final by deliberately getting the question wrong.

But in the real world, nobody has perfect knowledge. The other pair has a reasonable chance of just plain getting the question wrong. You have a chance of getting question 10 wrong. As a result it's almost certainly right to deliberately get the question wrong but in our fantasy world you have to go for it and hope.