Question

You play a game in which two coins are flipped. if both coins turn up tails, you win 1 point. how many points would you need to lose for each of the other outcomes so that the game is fair?

a: 1/4
b: 1/3
c: 1
d: 4/3

Answer 1

To make the game fair, you would need to lose $10 if one coin is heads and the other is tails, and $9 if both coins are heads.

Step-by-step explanation:

To make the game fair, we need to determine how many points you would need to lose for each outcome other than both coins turning up tails. Let's break it down:

1. If both coins turn up heads, you win $0 because the game rule specifies that you only win 1 point when both coins turn up tails.
2. If one coin turns up heads and the other turns up tails, you would need to lose $10 to make the game fair. This is because you would lose the 1 point you need to win when both coins turn up tails, plus an additional $9 since one of the coins is heads.
3. If both coins turn up heads, you would need to lose $9 to make the game fair. This is because you would lose the 1 point you need to win when both coins turn up tails, plus an additional $8 since both coins are heads.

Based on these calculations, the correct option would be d) 4/3.

Answer 2

To make the game fair, you would need to lose 1/3 point for each of the other outcomes.

Step-by-step explanation:

To determine how many points you would need to lose for each of the other outcomes in order for the game to be fair, we need to calculate the probabilities of each outcome.

There are four possible outcomes when flipping two coins: HH, HT, TH, and TT. In this game, you win 1 point if both coins turn up tails (TT), so the probability of winning is 1/4.

For the game to be fair, the expected value should be 0. This means that the probability of winning 1 point should be equal to the probability of losing a certain number of points for the other outcomes.

Let's suppose you lose x points for each of the other outcomes. The expected value of the game is then:

• P(TT) * 1 - P(HH) * x - P(HT) * x - P(TH) * x = 0
• 1/4 - P(HH) * x - P(HT) * x - P(TH) * x = 0

Since the probabilities of flipping two heads (HH), a head and a tail (HT), and a tail and a head (TH) are all 1/4, the equation becomes:

• 1/4 - (1/4 * x) - (1/4 * x) - (1/4 * x) = 0
• 1/4 - 3/4 * x = 0
• -3/4 * x = -1/4
• x = (1/4) / (3/4) = 1/3

Therefore, you would need to lose 1/3 point for each of the other outcomes in order for the game to be fair.