Question

1. tanya is considering playing a game at the fair. there are three different ones to choose from, and it costs $2 to play a game. the probabilities associated with the games are given in the table.

a. what is the expected value for playing each game?

b. if tanya decides she will play the game, which game should she choose? explain.

Explanation:

Step1: Calculate expected value for Game 1

The formula for expected value \(E(X)=\sum x_iP(x_i)\). For Game 1:
\(E(X_1)=(- 2)\times0.55+(1)\times0.20+(4)\times0.25\)
\(=-1.1 + 0.2+1\)
\(=0.1\)

Step2: Calculate expected value for Game 2

For Game 2:
\(E(X_2)=(-2)\times0.15+(1)\times0.35+(4)\times0.50\)
\(=-0.3+0.35 + 2\)
\(=2.05\)

Step3: Calculate expected value for Game 3

For Game 3:
\(E(X_3)=(-2)\times0.20+(1)\times0.60+(4)\times0.20\)
\(=-0.4 + 0.6+0.8\)
\(=1\)

Step4: Determine the best - game for Tanya

Since \(E(X_2)>E(X_3)>E(X_1)\)

Answer:

a. The expected value of Game 1 is \(0.1\), of Game 2 is \(2.05\), and of Game 3 is \(1\).
b. Tanya should choose Game 2. Because the expected value of Game 2 (\(E(X_2) = 2.05\)) is higher than the expected values of Game 1 (\(E(X_1)=0.1\)) and Game 3 (\(E(X_3) = 1\)). A higher expected value means, on average, she can expect to win more money per play.