Question

which of the following situations are fair?

ray is playing a game in which he rolls a six - sided number cube. if the outcome is six, he is paid $5. otherwise, he loses $1.

jack and mia both want the last cookie and neither will agree to share by splitting it. they ask a stranger passing by to flip a coin to decide who gets the cookie.

jason plays a game in which he has to pick a ball from a box of 10 balls, which contains 7 black balls and 3 white balls. he wins the game if he draws a white ball in one attempt.

to decide which citizens will be asked to participate in a county - wide poll, each citizen is assigned a number and the county uses a random number generator to determine the citizens who will be contacted.

jericho designs a game for a school carnival. the game consists of a box of 50 colored balls; 35 are violet, 10 are orange, and 5 are yellow. the player has to choose one ball from the box. the player wins $5 if it is a yellow ball, otherwise, the player wins nothing.

ryan designs a game where a prize wheel is split into five equal sections. four sections are red and one is green. if the wheel is spun and lands on a red section, the player loses $1. if the wheel lands on green section, the player wins $5.

Explanation:

To determine fairness, we check if the expected value (for games) or probability of each outcome is equal (for decisions):

1. Ray’s Game:

• Probability of 6: \( \frac{1}{6} \), payout: \( \$5 \).
• Probability of not 6: \( \frac{5}{6} \), payout: \( -\$1 \).
• Expected value: \( \frac{1}{6}(5) + \frac{5}{6}(-1) = \frac{5 - 5}{6} = 0 \). Fair.

2. Jack & Mia (Coin Flip):

A fair coin has \( P(\text{Heads}) = P(\text{Tails}) = \frac{1}{2} \). Equal chance for both. Fair.

3. Jason’s Ball Game:

• \( P(\text{White}) = \frac{3}{10} \), \( P(\text{Black}) = \frac{7}{10} \). Unequal probabilities. Unfair.

4. County Poll (Random Number):

Each citizen has an equal chance (random selection). Fair.

5. Jericho’s Ball Game:

• \( P(\text{Yellow}) = \frac{5}{50} = \frac{1}{10} \), \( P(\text{Not Yellow}) = \frac{45}{50} = \frac{9}{10} \). Unequal probabilities. Unfair.

6. Ryan’s Prize Wheel:

• \( P(\text{Red}) = \frac{4}{5} \), payout: \( -\$1 \).
• \( P(\text{Green}) = \frac{1}{5} \), payout: \( \$5 \).
• Expected value: \( \frac{4}{5}(-1) + \frac{1}{5}(5) = \frac{-4 + 5}{5} = \frac{1}{5}

eq 0 \). Unfair.

Answer:

• Ray is playing a game... (fair)
• Jack and Mia both want... (fair)
• To decide which citizens... (fair)