Consider a state lottery that has a weekly television show. On this show, a contestant receives the opportunity to win $1 million. The contestant picks from four hidden windows. Behind each is one of the following: $150,000, $200,000, $1 million, or a "stopper". Before beginning, the contestant is offered $100,000 to stop. Mathematically speaking, should the contestant take the $100,000?
To determine whether the contestant should take the $100,000 offer or continue with the game, we need to examine the expected value of each option.
An expected value is a weighted average of the possible outcomes, where each outcome is multiplied by its associated probability.
In this case, let's calculate the expected value of continuing the game:
1) Probability of picking $150,000: 1/4
2) Probability of picking $200,000: 1/4
3) Probability of picking $1 million: 1/4
4) Probability of picking the "stopper": 1/4
To calculate the expected value, we multiply the amount behind each window by its probability and sum them up:
Expected Value = ($150,000 * 1/4) + ($200,000 * 1/4) + ($1 million * 1/4) + ($0 * 1/4)
= $37,500 + $50,000 + $250,000 + $0
= $337,500
So, the expected value of continuing the game is $337,500.
Now, let's compare this to the $100,000 offer:
If the contestant accepts the $100,000, they receive a guaranteed amount.
Therefore, if the expected value of continuing the game ($337,500) is higher than the $100,000 offer, it would be mathematically better for the contestant to reject the offer and continue playing. However, if the expected value is lower than $100,000, it would be better to take the offer.
In this case, since the expected value of continuing the game ($337,500) is higher than the $100,000 offer, mathematically it would be better for the contestant to reject the offer and continue playing.