Question

You roll a die. If it comes up a 5 or 6, you win $1. If not, you get to roll again. If you get the second time 5 or 6, you win $10. If not, you lose.

a) Create a probability model for the amount you win.

b) Find the expected amount you'll win.

c) What should you be willing to pay to play this game?

Answer 1

The probability model includes winning $1 with a 2/6 chance, winning $10 with a 4/6 * 2/6 chance, and losing with a (4/6) * (4/6) chance. The expected value of this dice-rolling game is $1.78. You should be willing to pay up to $1.78 to play the game.

Step-by-step explanation:

You have a game where you roll a die, and if it lands on a 5 or 6, you win $1 on the first roll. If that doesn't happen, you get to roll again, and if you roll a 5 or 6 on the second try, you win $10; otherwise, you lose.

Probability Model

• Win $1: Probability is 2/6 because there are two winning numbers out of six possible numbers.
• Win $10: Probability is (4/6) * (2/6), because you must first lose (not roll a 5 or 6), which has a probability of 4/6, and then win on the second roll with a probability of 2/6.
• Lose: Probability is (4/6) * (4/6), which is the probability of not getting a 5 or 6 on both rolls.

Expected Winnings

The expected value is calculated as follows:

(2/6) * $1 + (4/6) * (2/6) * $10 - (4/6) * (4/6) * $0 = $1.78.
Cost to Play

You should be willing to pay up to the expected winnings to play the game, so in this case, up to $1.78.