Let's solve this problem step-by-step.
### Part a: Expected Value Calculation
The expected value for each game can be calculated using the formula:
[tex]\[ \text{Expected Value} = \sum (\text{Payout} \times \text{Probability}) \][/tex]
#### Game 1:
- Lose \[tex]$2 with probability 0.55
- Win \$[/tex]1 with probability 0.20
- Win \[tex]$4 with probability 0.25
Calculate the expected value:
\[
\text{Expected Value for Game 1} = (-2 \times 0.55) + (1 \times 0.20) + (4 \times 0.25)
\]
\[
= -1.1 + 0.2 + 1.0
\]
\[
= 0.10
\]
#### Game 2:
- Lose \$[/tex]2 with probability 0.15
- Win \[tex]$1 with probability 0.35
- Win \$[/tex]4 with probability 0.50
Calculate the expected value:
[tex]\[
\text{Expected Value for Game 2} = (-2 \times 0.15) + (1 \times 0.35) + (4 \times 0.50)
\][/tex]
[tex]\[
= -0.3 + 0.35 + 2.0
\][/tex]
[tex]\[
= 2.05
\][/tex]
#### Game 3:
- Lose \[tex]$2 with probability 0.20
- Win \$[/tex]1 with probability 0.60
- Win \[tex]$4 with probability 0.20
Calculate the expected value:
\[
\text{Expected Value for Game 3} = (-2 \times 0.20) + (1 \times 0.60) + (4 \times 0.20)
\]
\[
= -0.4 + 0.6 + 0.8
\]
\[
= 1.00
\]
### Summary of Expected Values
- Expected Value for Game 1: \$[/tex]0.10
- Expected Value for Game 2: \[tex]$2.05
- Expected Value for Game 3: \$[/tex]1.00
### Part b: Which Game Should Tanya Choose?
The expected value represents the average amount she expects to win per game in the long run. Among the three games, we compare the expected values:
- Game 1: \[tex]$0.10
- Game 2: \$[/tex]2.05
- Game 3: \[tex]$1.00
Since Game 2 has the highest expected value of \$[/tex]2.05, Tanya should choose Game 2 to maximize her expected winnings.
