Question

end of semester test
12
select all the correct answers.
which of the following situations are fair?
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 players loses $1. if the wheel lands on green section, the player wins $5.
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.
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.
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.

Explanation:

Step1: Analyze the county - wide poll situation

Random number generator gives equal probability to each citizen, so it is fair.

Step2: Analyze Jericho's game

Probability of getting a yellow ball is $\frac{5}{50}=\frac{1}{10}$. The pay - off structure may not be balanced with the probability, so it may not be fair.

Step3: Analyze Ryan's game

Probability of landing on red is $\frac{4}{5}$ and on green is $\frac{1}{5}$. Expected value: $E = \frac{4}{5}\times(- 1)+\frac{1}{5}\times5=\frac{-4 + 5}{5}=\frac{1}{5}>0$, not a fair - game in a strict sense as the expected value is non - zero.

Step4: Analyze Jason's game

Probability of drawing a white ball is $\frac{3}{10}$, no information about pay - offs to determine fairness.

Step5: Analyze Jack and Mia's situation

Coin flip has two equally likely outcomes (heads or tails), so it is fair.

Step6: Analyze Ray's game

Probability of getting a six is $\frac{1}{6}$, and probability of not getting a six is $\frac{5}{6}$. Expected value: $E=\frac{1}{6}\times5+\frac{5}{6}\times(-1)=\frac{5 - 5}{6}=0$, so it is fair.

Answer:

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.
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.
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.