if there 60% math major are women, 15 were random. what probility 12 women were are selected and what are the mean and variance. Show ne how to work the problems
To calculate the probability of selecting 12 women out of a group where 60% are math majors, we need to use the concept of probability. Here's how you can work out this problem:
Step 1: Find the probability of selecting a woman math major.
- Given that 60% of math majors are women, the probability of selecting a woman math major is 0.60.
Step 2: Calculate the probability of selecting 12 women math majors out of 15 random selections.
- Since the selections are random, we can use the concept of independent events and the binomial probability formula.
- The binomial probability formula is P(x) = C(n, x) * p^x * q^(n-x), where P(x) is the probability of getting x successes out of n trials, C(n, x) is the number of combinations of n things taken x at a time, p is the probability of success, and q is the probability of failure.
In this case, the number of trials (n) is 15, the number of successes (x) is 12, and the probability of selecting a woman math major (p) is 0.60. The probability of selecting a man math major (q) is 1 - 0.60 = 0.40.
Using the binomial probability formula, we can calculate the probability of selecting 12 women math majors out of 15 random selections:
P(12) = C(15, 12) * (0.60)^12 * (0.40)^(15-12)
= (15! / (12! * (15-12)!)) * (0.60)^12 * (0.40)^3
= (15 * 14 * 13 / (3 * 2 * 1)) * (0.60)^12 * (0.40)^3
≈ 0.2454
Therefore, the probability of selecting 12 women math majors out of 15 random selections is approximately 0.2454.
To calculate the mean and variance, we need to use the properties of the binomial distribution.
Mean (μ) = n * p
= 15 * 0.60
= 9
Variance (σ^2) = n * p * q
= 15 * 0.60 * 0.40
= 3.6
Therefore, the mean of the distribution is 9 and the variance is 3.6.