If 30% of students over the age of 18 smoke and 9 students are chosen at random. What is the probability that less than 2 of them smoke.
p(s) = probability smoke , p(d) = probability don't smoke
p(s) + p(d) = .3 + .7 = 1
< 2 means one or none
(d + s)^9 = d^9 + 9 d^8 s + ... + 9 d s^8 + s^9
the sum of the 1st two terms is the solution
... (.7)^9 + [9 * (.7)^8 * .3]
To find the probability that less than 2 of the 9 randomly chosen students smoke, we need to calculate the probability of both 0 and 1 student smoking.
Step 1: Calculate the probability of 0 students smoking:
Given that 30% of students over the age of 18 smoke, the probability of a student smoking is 0.30. Thus, the probability of a student not smoking is 1 - 0.30 = 0.70. Since we want to choose 0 students who smoke, we need to calculate the probability of not smoking for all 9 students. The probability of 0 students smoking can be calculated using the binomial distribution formula:
P(X = 0) = (nCr) * (p^x) * ((1-p)^(n-x))
where n = total number of trials (9 students), x = number of successes (0 students smoking), p = probability of success (probability of a student not smoking).
Using the formula:
P(X = 0) = (9C0) * (0.70^0) * (0.30^9)
= (1) * (1) * (0.000238)
= 0.000238
Step 2: Calculate the probability of 1 student smoking:
Similar to Step 1, we can use the binomial distribution formula to calculate the probability of exactly 1 student smoking:
P(X = 1) = (nCr) * (p^x) * ((1-p)^(n-x))
where n = total number of trials (9 students), x = number of successes (1 student smoking), p = probability of success (probability of a student not smoking).
Using the formula:
P(X = 1) = (9C1) * (0.70^1) * (0.30^8)
= (9) * (0.70) * (0.000653)
= 0.004579
Step 3: Calculate the probability that less than 2 students smoke:
To find the probability that less than 2 students smoke, we need to sum the probabilities from Step 1 and Step 2:
P(X < 2) = P(X = 0) + P(X = 1)
= 0.000238 + 0.004579
= 0.004817
Therefore, the probability that less than 2 of the 9 randomly chosen students smoke is approximately 0.004817, or 0.4817%.