Question

The student body of a large university consists of 60% female students. A random sample of 8 students is selected. What is the probability that among the students in the sample at least 6 are female?

Answer 1

Using the binomial distribution, it is found that there is a 0.3154 = 31.54% probability that among the students in the sample at least 6 are female.

What is the binomial distribution formula?

The formula is:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$C_{n,x} = \frac{n!}{x!(n-x)!}$

The parameters are:

• x is the number of successes.

• n is the number of trials.

• p is the probability of a success on a single trial.

In this problem:

• The student body of a large university consists of 60% female students, hence p = 0.6.

• A random sample of 8 students is selected, hence n = 8.

The probability that among the students in the sample at least 6 are female is given by:

$P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8)$

In which:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 6) = C_{8,6}.(0.6)^{6}.(0.4)^{2} = 0.2090$

$P(X = 7) = C_{8,7}.(0.6)^{7}.(0.4)^{1} = 0.0896$

$P(X = 8) = C_{8,8}.(0.6)^{8}.(0.4)^{0} = 0.0168$

Then:

$P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8) = 0.2090 + 0.0896 + 0.0168 = 0.3154$

0.3154 = 31.54% probability that among the students in the sample at least 6 are female.

More can be learned about the binomial distribution at https://brainly.com/question/24863377

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