To determine which plan has the higher payout in the long run, we'll calculate the expected value for each plan. The expected value of a plan is the sum of the products of each payout and its corresponding probability.
We start with Plan A, which has the following payouts and probabilities:
- [tex]$\$[/tex] -10,000[tex]$ with a probability of 0.61
- $[/tex]\[tex]$ 25,000$[/tex] with a probability of 0.15
- [tex]$\$[/tex] 55,000[tex]$ with a probability of 0.24
Calculating the expected value for Plan A:
\[
\text{Expected Value of Plan A} = (-\$[/tex]10,000 \times 0.61) + (\[tex]$25,000 \times 0.15) + (\$[/tex]55,000 \times 0.24)
\]
From previous calculations, the expected value of Plan A is:
[tex]\[
\$10,850
\][/tex]
Next, we consider Plan B, which has the following payouts and probabilities:
- [tex]$\$[/tex] -15,000[tex]$ with a probability of 0.16
- $[/tex]\[tex]$ -10,000$[/tex] with a probability of 0.43
- [tex]$\$[/tex] 30,000[tex]$ with a probability of 0.41
Calculating the expected value for Plan B:
\[
\text{Expected Value of Plan B} = (-\$[/tex]15,000 \times 0.16) + (-\[tex]$10,000 \times 0.43) + (\$[/tex]30,000 \times 0.41)
\]
From previous calculations, the expected value of Plan B is:
[tex]\[
\$5,600
\][/tex]
Comparing the two expected values:
- Plan A: \[tex]$10,850
- Plan B: \$[/tex]5,600
In the long run, Plan A has a higher payout with an expected value of \[tex]$10,850 compared to Plan B's expected value of \$[/tex]5,600. Therefore, Plan A is the better choice for a higher payout in the long run.
