# Suppose that 40% of all college students smoke cigarettes. A sample of 10 is selected randomly. What is the probability tha t

a half of the students smoke?

## To find the probability that exactly half of the students smoke, we can use the binomial probability formula. The formula is given by:

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

Where:

- P(X = k) is the probability that exactly k students smoke

- n is the sample size (10 students in this case)

- k is the number of students who smoke (in this case, k = 5)

- p is the probability of a student smoking (40% or 0.4)

- (n choose k) is the number of Ways to choose k students from n students, which can be calculated using the combination formula: (n! / (k! * (n - k)!))

Now, let's substitute the values into the formula:

P(X = 5) = (10 choose 5) * 0.4^5 * (1 - 0.4)^(10 - 5)

Calculating each term:

(10 choose 5) = (10! / (5! * (10 - 5)!))

= (10! / (5! * 5!))

= (10 * 9 * 8 * 7 * 6) / (5 * 4 * 3 * 2 * 1)

= 252

0.4^5 = 0.01024

(1 - 0.4)^(10 - 5) = 0.6^5 = 0.07776

Now, substituting these back into the formula:

P(X = 5) = 252 * 0.01024 * 0.07776

P(X = 5) ≈ 0.1999

Therefore, the probability that exactly half of the students smoke is approximately 0.1999 or 19.99%.

## To determine the probability that exactly half of the students smoke, we can use the binomial probability formula.

The binomial probability formula is:

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

Where:

- P(X = k) is the probability of exactly k successes

- n is the number of trials or sample size

- p is the probability of success on a single trial

- (n choose k) is the combination formula which calculates the number of ways to choose k items from a set of n items

In this case:

- n = 10 (the sample size)

- p = 0.4 (the probability of a student smoking)

- k = 5 (since half of the students would mean 5 out of 10)

Now we can plug these values into the formula and calculate the probability:

P(X = 5) = (10 choose 5) * 0.4^5 * (1 - 0.4)^(10 - 5)

Calculating the combination (10 choose 5):

(10 choose 5) = 10! / (5! * (10-5)!) = 252

Plugging in the values:

P(X = 5) = 252 * 0.4^5 * 0.6^5

Now we can simplify:

P(X = 5) = 252 * 0.01024 * 0.07776

Finally, calculating the result:

P(X = 5) = 0.1990656

Therefore, the probability that exactly half of the students smoke is approximately 0.199 or 19.9%.

## so prob that 5 of the 10 smoke

= C(10,5)(.4^5)(.6^5)

= appr .2

40% of college students smoke ???

Are those stats from the 50's ?