Answer :
Let's tackle Claire's investment decision and then move on to Tanya's game considerations.
### Claire's Investment Decision
#### Part a: Expected Value Calculation
To determine whether Claire should invest in the new business, we need to calculate the expected value (EV) of her investment. The expected value will give us the average outcome considering all possible scenarios and their probabilities.
The probabilities and outcomes are:
- Probability of losing [tex]\( \$10,000 \)[/tex]: [tex]\(0.2\)[/tex]
- Probability of breaking even ([tex]\(\$0\)[/tex] gain or loss): [tex]\(0.4\)[/tex]
- Probability of making [tex]\( \$5,000 \)[/tex]: [tex]\(0.3\)[/tex]
- Probability of making [tex]\( \$8,000 \)[/tex]: [tex]\(0.1\)[/tex]
The formula for the expected value is:
[tex]\[ EV = (P_{\text{lose}} \times \text{Loss}) + (P_{\text{break even}} \times \text{Break even}) + (P_{\text{profit1}} \times \text{Profit1}) + (P_{\text{profit2}} \times \text{Profit2}) \][/tex]
Plugging in the numbers:
[tex]\[ EV = (0.2 \times -10,000) + (0.4 \times 0) + (0.3 \times 5,000) + (0.1 \times 8,000) \][/tex]
[tex]\[ EV = -2,000 + 0 + 1,500 + 800 \][/tex]
[tex]\[ EV = 300 \][/tex]
Since the expected value is positive (\[tex]$300), it means that on average, Claire is expected to gain \$[/tex]300 by investing in the new business. Thus, Claire should invest in the business as it is expected to be profitable.
#### Part b: Break-even Calculation
Claire's initial investment is \[tex]$1,200. To determine how long it would take for her to recoup her initial investment, we use the expected annual profit calculated earlier (\$[/tex]300).
We calculate the number of years ( [tex]\( n \)[/tex] ) it would take for her to recover her initial investment using the formula:
[tex]\[ n = \frac{\text{Initial Investment}}{\text{Expected Annual Profit}} \][/tex]
Substituting the values:
[tex]\[ n = \frac{1,200}{300} \][/tex]
[tex]\[ n = 4 \][/tex]
It would take Claire 4 years to earn back her initial investment of \[tex]$1,200 if the expected value remains constant. ### Tanya's Game Decision Tanya has three games to choose from, each with its own set of probabilities and outcomes. Let's calculate the expected value for each game. #### Game 1 - Probability of losing \$[/tex]2: 0.55
- Probability of winning \[tex]$1: 0.20 - Probability of winning \$[/tex]3: 0.25
[tex]\[ EV_{\text{Game 1}} = (0.55 \times -2) + (0.20 \times 1) + (0.25 \times 3) \][/tex]
[tex]\[ EV_{\text{Game 1}} = -1.10 + 0.20 + 0.75 \][/tex]
[tex]\[ EV_{\text{Game 1}} = -0.15 \][/tex]
#### Game 2
- Probability of losing \[tex]$2: 0.15 - Probability of winning \$[/tex]1: 0.35
- Probability of winning \[tex]$3: 0.50 \[ EV_{\text{Game 2}} = (0.15 \times -2) + (0.35 \times 1) + (0.50 \times 3) \] \[ EV_{\text{Game 2}} = -0.30 + 0.35 + 1.50 \] \[ EV_{\text{Game 2}} = 1.55 \] #### Game 3 - Probability of losing \$[/tex]2: 0.20
- Probability of winning \[tex]$1: 0.60 - Probability of winning \$[/tex]2: 0.20
[tex]\[ EV_{\text{Game 3}} = (0.20 \times -2) + (0.60 \times 1) + (0.20 \times 2) \][/tex]
[tex]\[ EV_{\text{Game 3}} = -0.40 + 0.60 + 0.40 \][/tex]
[tex]\[ EV_{\text{Game 3}} = 0.60 \][/tex]
By comparing the expected values:
- EV for Game 1: -\[tex]$0.15 - EV for Game 2: \$[/tex]1.55
- EV for Game 3: \[tex]$0.60 Tanya should choose Game 2 since it has the highest expected value of \$[/tex]1.55, indicating the highest average win.
In summary, Claire should invest in the business, and it will take her 4 years to recoup her initial investment based on the expected value. Tanya, on the other hand, should play Game 2 at the fair to maximize her expected winnings.
### Claire's Investment Decision
#### Part a: Expected Value Calculation
To determine whether Claire should invest in the new business, we need to calculate the expected value (EV) of her investment. The expected value will give us the average outcome considering all possible scenarios and their probabilities.
The probabilities and outcomes are:
- Probability of losing [tex]\( \$10,000 \)[/tex]: [tex]\(0.2\)[/tex]
- Probability of breaking even ([tex]\(\$0\)[/tex] gain or loss): [tex]\(0.4\)[/tex]
- Probability of making [tex]\( \$5,000 \)[/tex]: [tex]\(0.3\)[/tex]
- Probability of making [tex]\( \$8,000 \)[/tex]: [tex]\(0.1\)[/tex]
The formula for the expected value is:
[tex]\[ EV = (P_{\text{lose}} \times \text{Loss}) + (P_{\text{break even}} \times \text{Break even}) + (P_{\text{profit1}} \times \text{Profit1}) + (P_{\text{profit2}} \times \text{Profit2}) \][/tex]
Plugging in the numbers:
[tex]\[ EV = (0.2 \times -10,000) + (0.4 \times 0) + (0.3 \times 5,000) + (0.1 \times 8,000) \][/tex]
[tex]\[ EV = -2,000 + 0 + 1,500 + 800 \][/tex]
[tex]\[ EV = 300 \][/tex]
Since the expected value is positive (\[tex]$300), it means that on average, Claire is expected to gain \$[/tex]300 by investing in the new business. Thus, Claire should invest in the business as it is expected to be profitable.
#### Part b: Break-even Calculation
Claire's initial investment is \[tex]$1,200. To determine how long it would take for her to recoup her initial investment, we use the expected annual profit calculated earlier (\$[/tex]300).
We calculate the number of years ( [tex]\( n \)[/tex] ) it would take for her to recover her initial investment using the formula:
[tex]\[ n = \frac{\text{Initial Investment}}{\text{Expected Annual Profit}} \][/tex]
Substituting the values:
[tex]\[ n = \frac{1,200}{300} \][/tex]
[tex]\[ n = 4 \][/tex]
It would take Claire 4 years to earn back her initial investment of \[tex]$1,200 if the expected value remains constant. ### Tanya's Game Decision Tanya has three games to choose from, each with its own set of probabilities and outcomes. Let's calculate the expected value for each game. #### Game 1 - Probability of losing \$[/tex]2: 0.55
- Probability of winning \[tex]$1: 0.20 - Probability of winning \$[/tex]3: 0.25
[tex]\[ EV_{\text{Game 1}} = (0.55 \times -2) + (0.20 \times 1) + (0.25 \times 3) \][/tex]
[tex]\[ EV_{\text{Game 1}} = -1.10 + 0.20 + 0.75 \][/tex]
[tex]\[ EV_{\text{Game 1}} = -0.15 \][/tex]
#### Game 2
- Probability of losing \[tex]$2: 0.15 - Probability of winning \$[/tex]1: 0.35
- Probability of winning \[tex]$3: 0.50 \[ EV_{\text{Game 2}} = (0.15 \times -2) + (0.35 \times 1) + (0.50 \times 3) \] \[ EV_{\text{Game 2}} = -0.30 + 0.35 + 1.50 \] \[ EV_{\text{Game 2}} = 1.55 \] #### Game 3 - Probability of losing \$[/tex]2: 0.20
- Probability of winning \[tex]$1: 0.60 - Probability of winning \$[/tex]2: 0.20
[tex]\[ EV_{\text{Game 3}} = (0.20 \times -2) + (0.60 \times 1) + (0.20 \times 2) \][/tex]
[tex]\[ EV_{\text{Game 3}} = -0.40 + 0.60 + 0.40 \][/tex]
[tex]\[ EV_{\text{Game 3}} = 0.60 \][/tex]
By comparing the expected values:
- EV for Game 1: -\[tex]$0.15 - EV for Game 2: \$[/tex]1.55
- EV for Game 3: \[tex]$0.60 Tanya should choose Game 2 since it has the highest expected value of \$[/tex]1.55, indicating the highest average win.
In summary, Claire should invest in the business, and it will take her 4 years to recoup her initial investment based on the expected value. Tanya, on the other hand, should play Game 2 at the fair to maximize her expected winnings.
