Question

1 . Individual Problems 17-2

You're a contestant on a TV game show. In the final round of the game, if contestants answer a question correctly, they will increase their current winnings of $2 million to $3 million. If they are wrong, their prize is decreased to $1,000,000. You believe you have a 25% chance of answering the question correctly.

Ignoring your current winnings, your expected payoff from playing the final round of the game show is__________. Given that this is (Postive or Negative), you (Should, or Should not)   play the final round of the game. (Hint: Enter a negative sign if the expected payoff is negative.)

The lowest probability of a correct guess that would make the guessing in the final round profitable (in expected value) is (55.00, or 49.50, or 30.00, or 50.00) . (Hint: At what probability does playing the final round yield an expected value of zero?)

Answer 1

Solution:

In case of correct answer, the winnings increase by ($3 million - $2 million =) $1 million, and in case of incorrect answer, the winnings decrease by ($2 million - $1 million =) $1 million. With 25% chance of answering the question correctly, expected payoff = probability of answering correctly*earning from correct answer + probability of answering incorrectly*earning from incorrect answer

So, expected payoff = 0.25*1 million + (1 - 0.25)*(-1 million)

Expected payoff = 250000 - 750000 = -$500,000

Given that the expected payoff is negative, you should not play the final round of the game.

Let's denote the probability of answering correctly by p, so expected payoff = p*1 million + (1 - p)*(-1 million)

Expected payoff = p*1 million - 1 million + p*1 million

Expected payoff = 2p*1 million - 1 million

To make the final round profitable, expected payoff >= 0

2p*1 million - 1 million >= 0

p >= 1/2 or 0.50

So, lowest probability of correct guess is 50.00%.