Find the indicated binomial probabilities. Round to the nearest 3 decimal places.

In a local college, 20% of the math majors are women. Ten math majors are chosen at random.

1) What is the probability that exactly 2 are women?

2) What is the probability that 2 or less women are selected?

3) What is the probability that no women are selected?

4) Find the mean u

Here is the binomial probability function:

P(x) = (nCx)(p^x)[q^(n-x)]
n = 10
p = .20
q = .80 (q = 1 - p)

For 1), find P(2).
For 2), find P(0) + P(1) + P(2).
For 3), find P(0).
For 4), mean = np

I hope this will help get you started.

To find the indicated binomial probabilities, we use the binomial probability formula:

P(x) = (nCx) * p^x * q^(n-x)

where:
P(x) is the probability of getting exactly x successes
n is the total number of trials (in this case, the number of math majors chosen)
x is the number of successes (in this case, the number of women chosen)
p is the probability of success in each trial (the probability of choosing a woman)
q is the probability of failure in each trial (1 - p)

First, let's find the values of p and q:
p = 0.20 (probability of choosing a woman)
q = 1 - p = 0.80 (probability of choosing a man)

1) To find the probability that exactly 2 math majors are women:
P(2) = (10C2) * (0.20^2) * (0.80^(10-2))
= (10! / (2!(10-2)!)) * (0.20^2) * (0.80^8)
= (10! / (2!8!)) * (0.04) * (0.1677)
= (10 * 9 / (2 * 1)) * (0.04) * (0.1677)
= 45 * 0.04 * 0.1677
≈ 0.302 (rounded to 3 decimal places)

2) To find the probability that 2 or less math majors are women:
P(0) = (10C0) * (0.20^0) * (0.80^10)
= (10! / (0!(10-0)!)) * (0.20^0) * (0.80^10)
= (10! / (1!9!)) * (1) * (0.1074)
= (10 * 1 / (1 * 1)) * (1) * (0.1074)
= 10 * 1 * 0.1074
= 1.074

P(1) = (10C1) * (0.20^1) * (0.80^(10-1))
= (10! / (1!(10-1)!)) * (0.20^1) * (0.80^9)
= (10! / (1!9!)) * (0.20) * (0.1074)
= (10 * 1 / (1 * 1)) * (0.20) * (0.1074)
≈ 0.268

P(2) = (calculated earlier) ≈ 0.302

P(2 or less) = P(0) + P(1) + P(2)
≈ 1.074 + 0.268 + 0.302
≈ 1.644 (rounded to 3 decimal places)

3) To find the probability that no math majors are women:
P(0) = (calculated earlier) = 1.074

4) The mean (μ) of a binomial distribution is given by the formula:
μ = n * p

where n is the total number of trials and p is the probability of success in each trial.

In this case, n = 10 (the number of math majors chosen) and p = 0.20 (the probability of choosing a woman):
μ = 10 * 0.20
= 2

Therefore, the mean is 2.