To find the expected value of the winnings from a game with a given payout probability distribution, you apply the formula for the expected value of a discrete random variable.
The formula for the expected value [tex]\(E(X)\)[/tex] is:
[tex]\[ E(X) = \sum_{i=1}^{n} x_i \cdot p(x_i) \][/tex]
where:
- [tex]\(x_i\)[/tex] are the payout values,
- [tex]\(p(x_i)\)[/tex] are the probabilities associated with those payouts,
- and [tex]\(n\)[/tex] is the number of different payouts.
Using the given data:
- Payouts: [tex]\(0, 2, 4, 6, 8\)[/tex]
- Probabilities: [tex]\(0.36, 0.06, 0.33, 0.08, 0.17\)[/tex]
We calculate the expected value step by step:
1. Multiply each payout by its corresponding probability:
- [tex]\(0 \times 0.36 = 0\)[/tex]
- [tex]\(2 \times 0.06 = 0.12\)[/tex]
- [tex]\(4 \times 0.33 = 1.32\)[/tex]
- [tex]\(6 \times 0.08 = 0.48\)[/tex]
- [tex]\(8 \times 0.17 = 1.36\)[/tex]
2. Sum these products:
[tex]\[
0 + 0.12 + 1.32 + 0.48 + 1.36 = 3.2800000000000002
\][/tex]
Thus, the expected value of the winnings from the game is:
[tex]\[ \boxed{3.28} \][/tex]
Note: In calculations, you might often get results with many decimal places, like [tex]\(3.2800000000000002\)[/tex]. However, for practical purposes, it is typically rounded to a reasonable number of decimal places, in this case, two decimal places.
