if 60 % of all women are employed outside the home find the probability that in a sample of 20 women at least 10 are employed

To find the probability that in a sample of 20 women at least 10 are employed, we need to use the concept of binomial probability.

First, let's calculate the probability that exactly 10 women are employed. We can use the binomial probability formula:

P(X=k) = (n choose k) * p^k * (1-p)^(n-k)

Where:
- P(X=k) is the probability of getting exactly k successes (in this case, having k employed women)
- n is the total number of trials (in this case, the sample size of 20 women)
- k is the number of successful outcomes (in this case, 10 employed women)
- p is the probability of success (in this case, the probability of a woman being employed, which is 60% or 0.6)
- (n choose k) denotes the binomial coefficient, which is calculated as n! / (k! * (n-k)!), representing the number of ways to choose k successes out of n trials.

Using these values, we can calculate P(X=10):

P(X=10) = (20 choose 10) * (0.6)^10 * (1-0.6)^(20-10)

Calculating this gives us the probability of exactly 10 women being employed in the sample of 20 women.

Next, we need to calculate the probabilities for the remaining scenarios (11, 12, 13, ..., 20 employed women) and sum them up to find the probability that at least 10 women are employed. This can be done by repeating the calculation above for each value of k (11 to 20) and summing up the probabilities.

Alternatively, we can use a spreadsheet software like Microsoft Excel or Google Sheets to easily calculate these probabilities. By using the BINOM.DIST function, we can calculate the binomial probabilities for different values of k and sum them up to find the desired probability.

To use the BINOM.DIST function in Excel, you can use the following formula:

=1 - BINOM.DIST(9, 20, 0.6, TRUE)

This formula calculates the probability of having less than or equal to 9 employed women in a sample of 20 women and then subtracts it from 1 to get the probability of having at least 10 employed women.

Therefore, the probability that in a sample of 20 women at least 10 are employed can be found using the binomial probability formula or by using spreadsheet software like Excel.