To find the expected value of the winnings from the game, we need to use the formula for expected value in probability theory. The expected value [tex]\( E(X) \)[/tex] is calculated by multiplying each possible outcome by its probability and then summing all these products. Here's the step-by-step solution:
1. Identify the possible payouts and their associated probabilities:
- Payouts are [tex]\(\$ 2, \$ 4, \$ 6, \$ 8, \$ 10\)[/tex]
- Probabilities are [tex]\(0.36, 0.04, 0.12, 0.20, 0.28\)[/tex] respectively.
2. Multiply each payout by its probability:
- For [tex]\(\$ 2\)[/tex]: [tex]\(2 \times 0.36 = 0.72\)[/tex]
- For [tex]\(\$ 4\)[/tex]: [tex]\(4 \times 0.04 = 0.16\)[/tex]
- For [tex]\(\$ 6\)[/tex]: [tex]\(6 \times 0.12 = 0.72\)[/tex]
- For [tex]\(\$ 8\)[/tex]: [tex]\(8 \times 0.20 = 1.60\)[/tex]
- For [tex]\(\$ 10\)[/tex]: [tex]\(10 \times 0.28 = 2.80\)[/tex]
3. Sum all the results from step 2 to find the expected value:
[tex]\[
E(X) = 0.72 + 0.16 + 0.72 + 1.60 + 2.80 = 6.00
\][/tex]
So, the expected value of the winnings from the game is \$6.00.
