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Tanya is considering playing a game at the fair. There are three different games to choose from, and it costs [tex]$\$[/tex]2[tex]$ to play a game. The probabilities associated with the games are given in the table.

\begin{tabular}{l|l|l|l|}
\hline \multicolumn{1}{c}{ } & \multicolumn{1}{c}{Lose \$[/tex]2} & \multicolumn{1}{c}{Win \[tex]$1} & \multicolumn{1}{c}{Win \$[/tex]4} \\
\hline Game 1 & 0.55 & 0.20 & 0.25 \\
\hline Game 2 & 0.15 & 0.35 & 0.50 \\
\hline Game 3 & 0.20 & 0.60 & 0.20 \\
\hline
\end{tabular}

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.

Sagot :

GinnyAnsweridn
Let's break this problem into parts and solve it step-by-step.

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

To calculate the expected value for each game, we use the formula for expected value:
[tex]\[ \text{Expected Value} = \sum (\text{probability} \times \text{payoff}) \][/tex]

We'll calculate this for each game, using the probabilities and the payoffs provided.

Game 1:

- Probability of losing [tex]\(-\$2\)[/tex]: [tex]\(0.55\)[/tex]
- Probability of winning [tex]\(\$1\)[/tex]: [tex]\(0.20\)[/tex]
- Probability of winning [tex]\(\$4\)[/tex]: [tex]\(0.25\)[/tex]

[tex]\[ \text{Expected Value of Game 1} = (0.55 \times -2) + (0.20 \times 1) + (0.25 \times 4) \][/tex]

Game 2:

- Probability of losing [tex]\(-\$2\)[/tex]: [tex]\(0.15\)[/tex]
- Probability of winning [tex]\(\$1\)[/tex]: [tex]\(0.35\)[/tex]
- Probability of winning [tex]\(\$4\)[/tex]: [tex]\(0.50\)[/tex]

[tex]\[ \text{Expected Value of Game 2} = (0.15 \times -2) + (0.35 \times 1) + (0.50 \times 4) \][/tex]

Game 3:

- Probability of losing [tex]\(-\$2\)[/tex]: [tex]\(0.20\)[/tex]
- Probability of winning [tex]\(\$1\)[/tex]: [tex]\(0.60\)[/tex]
- Probability of winning [tex]\(\$4\)[/tex]: [tex]\(0.20\)[/tex]

[tex]\[ \text{Expected Value of Game 3} = (0.20 \times -2) + (0.60 \times 1) + (0.20 \times 4) \][/tex]

Now, evaluating these expressions:

For Game 1:
[tex]\[ (0.55 \times -2) + (0.20 \times 1) + (0.25 \times 4) = -1.1 + 0.2 + 1.0 = 0.1 \][/tex]
So, the expected value for Game 1 is [tex]\(0.1\)[/tex].

For Game 2:
[tex]\[ (0.15 \times -2) + (0.35 \times 1) + (0.50 \times 4) = -0.3 + 0.35 + 2.0 = 2.05 \][/tex]
So, the expected value for Game 2 is [tex]\(2.05\)[/tex].

For Game 3:
[tex]\[ (0.20 \times -2) + (0.60 \times 1) + (0.20 \times 4) = -0.4 + 0.6 + 0.8 = 1.0 \][/tex]
So, the expected value for Game 3 is [tex]\(1.0\)[/tex].

In summary, the expected values are:
- Game 1: [tex]\(0.1\)[/tex]
- Game 2: [tex]\(2.05\)[/tex]
- Game 3: [tex]\(1.0\)[/tex]

b. If Tanya decides she will play the game, which game should she choose? Explain.

To determine which game Tanya should choose, we look at the expected values calculated:
- Expected value of Game 1: [tex]\(0.1\)[/tex]
- Expected value of Game 2: [tex]\(2.05\)[/tex]
- Expected value of Game 3: [tex]\(1.0\)[/tex]

Since Tanya wants to maximize her expected winnings, she should choose the game with the highest expected value. Among the three games, Game 2 has the highest expected value of [tex]\(2.05\)[/tex].

Therefore, Tanya should choose Game 2.
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