Answer:
A.) P(X ≥ 7) = 1 - 0.9991356416 = **0.0008643584**
B.) E(X) = 10 * 0.2 = 2 prizes
Explanation:
a) Probability of winning at least seven prizes
This can be calculated using the binomial distribution formula:
P(X ≥ 7) = 1 - P(X ≤ 6)
where:
X is the number of prizes won
P(X ≥ 7) is the probability of winning at least seven prizes
P(X ≤ 6) is the probability of winning six or fewer prizes
The probability of winning a prize on each cup is 0.2, and the probability of not winning a prize is 0.8. We can use these values to calculate the probability of winning six or fewer prizes:
P(X ≤ 6) = Σ_(k=0)^(6) (10C_k) * 0.2^k * 0.8^(10-k) = 0.9991356416
Therefore, the probability of winning at least seven prizes is:
P(X ≥ 7) = 1 - 0.9991356416 = **0.0008643584**
b) Expected number of prizes
The expected number of prizes can be calculated using the following formula:
E(X) = np
where:
E(X) is the expected number of prizes
n is the number of cups of coffee purchased (10 in this case)
p is the probability of winning a prize on each cup (0.2 in this case)
Therefore, the expected number of prizes is:
E(X) = 10 * 0.2 = **2 prizes**
Therefore, the probability of winning at least seven prizes is 0.0008643584, and the expected number of prizes is 2.