Answer :
Certainly! Let's solve this step-by-step.
### Part (a) Which offer has the greater expected value?
First, we need to understand the concept of expected value. The expected value (EV) is a measure of the center of a probability distribution and is calculated by multiplying each possible outcome by the probability of that outcome and then summing these products.
Mom's Plan:
- Probability of drawing a red card: There are 26 red cards out of 52 cards in a standard deck of cards.
[tex]\[ P(\text{Red card}) = \frac{26}{52} = \frac{1}{2} = 0.5 \][/tex]
- Probability of drawing a black card: There are also 26 black cards out of 52 cards in a standard deck.
[tex]\[ P(\text{Black card}) = \frac{26}{52} = \frac{1}{2} = 0.5 \][/tex]
- Rewards:
- If she picks a red card, she gets [tex]\( \$15 \)[/tex].
- If she picks a black card, she gets [tex]\( \$0 \)[/tex].
The expected value for Mom's plan [tex]\( \text{EV}_{\text{Mom}} \)[/tex] is calculated as follows:
[tex]\[ \text{EV}_{\text{Mom}} = (\text{Probability of Red card} \times \text{Reward for Red card}) + (\text{Probability of Black card} \times \text{Reward for Black card}) \][/tex]
[tex]\[ \text{EV}_{\text{Mom}} = (0.5 \times 15) + (0.5 \times 0) = 7.5 + 0 = \$7.5 \][/tex]
Dad's Plan:
- Probability of each roll on a fair 6-sided die: Each face (1 through 6) has an equal probability of showing up.
[tex]\[ P(\text{each face}) = \frac{1}{6} \approx 0.1667 \][/tex]
- Rewards based on die roll: The reward is [tex]\( \$3 \)[/tex] per dot.
- If the die shows 1, reward is [tex]\( 1 \times 3 = \$3 \)[/tex].
- If the die shows 2, reward is [tex]\( 2 \times 3 = \$6 \)[/tex].
- If the die shows 3, reward is [tex]\( 3 \times 3 = \$9 \)[/tex].
- If the die shows 4, reward is [tex]\( 4 \times 3 = \$12 \)[/tex].
- If the die shows 5, reward is [tex]\( 5 \times 3 = \$15 \)[/tex].
- If the die shows 6, reward is [tex]\( 6 \times 3 = \$18 \)[/tex].
The expected value for Dad's plan [tex]\( \text{EV}_{\text{Dad}} \)[/tex] is calculated by summing the products of each outcome and its probability:
[tex]\[ \text{EV}_{\text{Dad}} = \left(\frac{1}{6} \times 3\right) + \left(\frac{1}{6} \times 6\right) + \left(\frac{1}{6} \times 9\right) + \left(\frac{1}{6} \times 12\right) + \left(\frac{1}{6} \times 15\right) + \left(\frac{1}{6} \times 18\right) \][/tex]
[tex]\[ \text{EV}_{\text{Dad}} = \frac{1}{6} \times (3 + 6 + 9 + 12 + 15 + 18) = \frac{1}{6} \times 63 = 10.5 \text{ dollars} \][/tex]
### Part (b) If you were Alicia, which plan would you pick? Why? Justify your answer.
Given the expected values calculated:
- Mom's plan expected value: \[tex]$7.5 - Dad's plan expected value: \$[/tex]10.5
Alicia should pick the plan with the greater expected value. In this case, Dad's plan has a higher expected value of [tex]\(10.5\)[/tex] dollars compared to Mom's plan's expected value of [tex]\(7.5\)[/tex] dollars.
Justification:
Expected value is a key factor in decision-making, especially under uncertainty. It represents the average outcome if the experiment (drawing a card or rolling a die) is repeated many times. Since Dad's plan offers a higher expected value, it means that on average, Alicia will receive more money by choosing Dad's plan over Mom's plan. Hence, the rational choice for Alicia would be to choose Dad’s plan.
### Part (a) Which offer has the greater expected value?
First, we need to understand the concept of expected value. The expected value (EV) is a measure of the center of a probability distribution and is calculated by multiplying each possible outcome by the probability of that outcome and then summing these products.
Mom's Plan:
- Probability of drawing a red card: There are 26 red cards out of 52 cards in a standard deck of cards.
[tex]\[ P(\text{Red card}) = \frac{26}{52} = \frac{1}{2} = 0.5 \][/tex]
- Probability of drawing a black card: There are also 26 black cards out of 52 cards in a standard deck.
[tex]\[ P(\text{Black card}) = \frac{26}{52} = \frac{1}{2} = 0.5 \][/tex]
- Rewards:
- If she picks a red card, she gets [tex]\( \$15 \)[/tex].
- If she picks a black card, she gets [tex]\( \$0 \)[/tex].
The expected value for Mom's plan [tex]\( \text{EV}_{\text{Mom}} \)[/tex] is calculated as follows:
[tex]\[ \text{EV}_{\text{Mom}} = (\text{Probability of Red card} \times \text{Reward for Red card}) + (\text{Probability of Black card} \times \text{Reward for Black card}) \][/tex]
[tex]\[ \text{EV}_{\text{Mom}} = (0.5 \times 15) + (0.5 \times 0) = 7.5 + 0 = \$7.5 \][/tex]
Dad's Plan:
- Probability of each roll on a fair 6-sided die: Each face (1 through 6) has an equal probability of showing up.
[tex]\[ P(\text{each face}) = \frac{1}{6} \approx 0.1667 \][/tex]
- Rewards based on die roll: The reward is [tex]\( \$3 \)[/tex] per dot.
- If the die shows 1, reward is [tex]\( 1 \times 3 = \$3 \)[/tex].
- If the die shows 2, reward is [tex]\( 2 \times 3 = \$6 \)[/tex].
- If the die shows 3, reward is [tex]\( 3 \times 3 = \$9 \)[/tex].
- If the die shows 4, reward is [tex]\( 4 \times 3 = \$12 \)[/tex].
- If the die shows 5, reward is [tex]\( 5 \times 3 = \$15 \)[/tex].
- If the die shows 6, reward is [tex]\( 6 \times 3 = \$18 \)[/tex].
The expected value for Dad's plan [tex]\( \text{EV}_{\text{Dad}} \)[/tex] is calculated by summing the products of each outcome and its probability:
[tex]\[ \text{EV}_{\text{Dad}} = \left(\frac{1}{6} \times 3\right) + \left(\frac{1}{6} \times 6\right) + \left(\frac{1}{6} \times 9\right) + \left(\frac{1}{6} \times 12\right) + \left(\frac{1}{6} \times 15\right) + \left(\frac{1}{6} \times 18\right) \][/tex]
[tex]\[ \text{EV}_{\text{Dad}} = \frac{1}{6} \times (3 + 6 + 9 + 12 + 15 + 18) = \frac{1}{6} \times 63 = 10.5 \text{ dollars} \][/tex]
### Part (b) If you were Alicia, which plan would you pick? Why? Justify your answer.
Given the expected values calculated:
- Mom's plan expected value: \[tex]$7.5 - Dad's plan expected value: \$[/tex]10.5
Alicia should pick the plan with the greater expected value. In this case, Dad's plan has a higher expected value of [tex]\(10.5\)[/tex] dollars compared to Mom's plan's expected value of [tex]\(7.5\)[/tex] dollars.
Justification:
Expected value is a key factor in decision-making, especially under uncertainty. It represents the average outcome if the experiment (drawing a card or rolling a die) is repeated many times. Since Dad's plan offers a higher expected value, it means that on average, Alicia will receive more money by choosing Dad's plan over Mom's plan. Hence, the rational choice for Alicia would be to choose Dad’s plan.
