Answer:
To find the binomial probability of getting at least 7 women when 10 people are picked from a pool with an equal chance of selecting either gender (50% chance for each), you can use the binomial probability formula.
The binomial probability formula is:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of getting exactly k women.
n is the number of trials (in this case, the number of people picked, which is 10).
k is the number of successful trials (in this case, the number of women you want to pick).
p is the probability of success on a single trial (in this case, the probability of picking a woman, which is 0.5 or 50%).
(n choose k) represents the binomial coefficient, which can be calculated as C(n, k) = n! / (k! * (n - k)!).
Now, you want to find the probability of getting at least 7 women, so you need to calculate the probability of getting exactly 7 women, exactly 8 women, exactly 9 women, and exactly 10 women, and then sum these probabilities.
P(X ≥ 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
Let's calculate each of these probabilities and then sum them:
P(X = 7) = C(10, 7) * (0.5)^7 * (0.5)^(10-7) = 120 * (0.5)^7 * (0.5)^3 = 120 * (0.5)^10
P(X = 8) = C(10, 8) * (0.5)^8 * (0.5)^(10-8) = 45 * (0.5)^8 * (0.5)^2 = 45 * (0.5)^10
P(X = 9) = C(10, 9) * (0.5)^9 * (0.5)^(10-9) = 10 * (0.5)^9 * (0.5)^1 = 10 * (0.5)^10
P(X = 10) = C(10, 10) * (0.5)^10 * (0.5)^(10-10) = (0.5)^10
Now, sum these probabilities:
P(X ≥ 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) = 120 * (0.5)^10 + 45 * (0.5)^10 + 10 * (0.5)^10 + (0.5)^10
P(X ≥ 7) = (120 + 45 + 10 + 1) * (0.5)^10 = 176 * (0.5)^10 = 0.172
So, the binomial probability of getting at least 7 women when 10 people are picked is approximately 0.172.
The answer is B. 0.172.