Answer: Tuesday Teaser #1
Despite being very similar to the Monty Hall problem, the odds in the Deal or No Deal scenario are different and actually work out to there being an equal chance that the player’s box contains the top prize. To see why this is the case, simply consider why both games are different: In the Monty Hall problem, we can never reach the stage where the top prize cannot be won, but in Deal or No Deal this is a definite, indeed likely, possibility.
We can do better than this and calculate the odds using conditional probability. Conditional probability allows us to use known events/information to update the probabilities of certain outcomes occurring. In this example, we would use it to say: “If the top prize is not in my box, there is only a very small chance of choosing 20 boxes without eliminating the top prize. Therefore, if this were to occur, it makes it more likely that the top prize was in my box to start with (where it couldn’t have been eliminated).”
We can formalise this with a little simple maths - we say the probability of event A happening given that event B has occurred if the probability of both events happening [written P(A∩B)] divided by the probability of event B happening [written P(B)]. Let’s say event A is the event that the top prize is in my box and that event B is the event that the top prize has not been eliminated at the last stage of the game. Now for a little maths:
Event B can occur in two ways:
Either the top prize is in my box at the beginning (1/22 chance) or it’s not and I manage to miss it (the chance of this is the chance that it is not in my box at the beginning multiplied by the chance that I miss it each time I choose a box to eliminate, i.e. 21/22 x 20/21 x 19/20 x … x 1/2 = 1/22). So event B has a 2/22 (or 1/11) chance.
Now it is clear that if my box contains the top prize, it will always be left at the end, so the probability of both events, P(A∩B), is just the probability that the top prize was in my box at the beginning (1/22 chance).
Applying the formula, we see that there is a 50-50 chance that my box contains the top prize given that it remains uneliminated at the last stage.
