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

5) Find the variance o2

Prb(woman) = .2

prb(man) = .8

1)choosing 10 ...
prob(exactly 2 women) = C(10,2)(.2)^2(.8)^8
= .302

etc.

Also look at the reply to the next post, I think it is your post.

To find the binomial probabilities, we need to use the binomial probability formula. The binomial probability formula is:

P(x) = (nCx) * p^x * (1-p)^(n-x)

Where:
- P(x) is the probability of getting exactly x successes
- n is the total number of trials or observations
- x is the number of successes we are interested in
- p is the probability of success in a single trial or observation
- (nCx) is the number of combinations of n trials taken x at a time, which can be calculated using the combination formula: (nCx) = n! / (x! * (n-x)!)

Now let's answer the specific questions using the binomial probability formula:

1) What is the probability that exactly 2 are women?
- Here, n = 10 (total number of math majors chosen at random), x = 2 (exactly 2 women), and p = 0.20 (probability of a math major being a woman).
- Calculate P(2) = (10C2) * (0.20^2) * (1-0.20)^(10-2) = 45 * 0.04 * 0.8^8 ≈ 0.302 (rounded to 3 decimal places).

2) What is the probability that 2 or less women are selected?
- Calculate P(0) + P(1) + P(2):
P(0) = (10C0) * (0.20^0) * (1-0.20)^(10-0) = 1 * 1 * 0.8^10 ≈ 0.107
P(1) = (10C1) * (0.20^1) * (1-0.20)^(10-1) = 10 * 0.20 * 0.8^9 ≈ 0.268
P(2) ≈ 0.302 (from the previous question)
- Calculate P(2 or less) = P(0) + P(1) + P(2) ≈ 0.107 + 0.268 + 0.302 ≈ 0.677

3) What is the probability that no women are selected?
- Calculate P(0) = (10C0) * (0.20^0) * (1-0.20)^(10-0) = 1 * 1 * 0.8^10 ≈ 0.107 (same as P(0) in the previous question)

4) Finding the mean (μ):
- 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.
- Calculate the mean μ = 10 * 0.20 = 2

5) Finding the variance (σ^2):
- The variance (σ^2) of a binomial distribution is given by the formula σ^2 = n * p * (1-p).
- Calculate the variance σ^2 = 10 * 0.20 * (1-0.20) = 1.6

So the mean (μ) is 2 and the variance (σ^2) is 1.6.