Question

17-2 Game Show Uncertainty. In the final round of a TV game show, contestants have a chance to increase their current winnings of $1 million to $2 million. If they are wrong, their prize is decreased to $500,000. A contestant thinks his guess will be right 50% of the time. Should he play? What is the lowest probability of a correct guess that would make playing profitable?

A) 11.5%
B) 32.5%
C) 50%
D) 85%

Answer 1

The contestant should play as the expected value is greater than their current winnings. The lowest probability of a correct guess that would make playing profitable is 32.5%.

Step-by-step explanation:

To determine if the contestant should play, we can calculate the expected value of playing. The expected value is calculated by multiplying the possible outcomes by their probabilities and then summing them up.

Let's assume the contestant plays once:

• If their guess is right (probability of 50%), they will win $2 million.
• If their guess is wrong (probability of 50%), their prize will decrease to $500,000.

Now, let's calculate the expected value:

Expected Value = (0.5 * $2,000,000) + (0.5 * $500,000) = $1,250,000

Based on the expected value calculation, the contestant should play since the expected value is greater than their current winnings of $1 million. Therefore, playing the final round would be profitable.

To find the lowest probability of a correct guess that would make playing profitable, we can set up an equation:

Expected Value = (Probability of Correct Guess * $2,000,000) + ((1 - Probability of Correct Guess) * $500,000)

By solving this equation, we find that the lowest probability of a correct guess that would make playing profitable is 32.5% (Option B).