According to the Smart magazine, 70% of smokers started smoking before the age of 18. Suppose 10 smokers are randomly selected and the number of smokers who started to smoke before the age of 18 is recorded.
a) explain why this is binomial experiment.
b) find the probability that exactly 7 of them started smoking before the age of 18.
c) find the probability that at least 7 of them started smoking before the age of 18.
d) find the probability that fewer than 4 of them started smoking before the age of 18.
e) find the probability that between 8 and 10 (inclusive) of them started smoking before the age of 18.
a) This is a binomial experiment because it satisfies the four criteria for a binomial experiment:
1) There are a fixed number of trials: The experiment involves selecting 10 smokers, and the number of trials is determined in advance as 10.
2) Each trial is independent: The choice of whether or not a smoker started smoking before the age of 18 does not depend on the choices of other smokers. Each trial is independent of each other.
3) There are only two possible outcomes: The outcomes are either the smoker started smoking before the age of 18 or not. It can be represented as a success (starting to smoke before 18) or failure (not starting to smoke before 18).
4) The probability of success is constant: The probability of a smoker starting to smoke before 18 is consistently given as 70%.
b) To find the probability that exactly 7 of them started smoking before the age of 18, we can use the binomial probability formula. The formula is:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
In this case, n = 10 (the number of trials), k = 7 (the number of successes), and p = 0.70 (the probability of success). Plugging in these values into the formula, we get:
P(X = 7) = (10 choose 7) * (0.70)^7 * (1 - 0.70)^(10 - 7)
c) To find the probability that at least 7 of them started smoking before the age of 18, we need to calculate the probabilities of 7, 8, 9, and 10 successes separately, and then add them together.
P(X >= 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
Calculate each of these probabilities using the formula mentioned in part b).
d) To find the probability that fewer than 4 of them started smoking before the age of 18, we need to calculate the probabilities of 0, 1, 2, and 3 successes separately, and then add them together.
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
Calculate each of these probabilities using the formula mentioned in part b).
e) To find the probability that between 8 and 10 (inclusive) of them started smoking before the age of 18, we need to calculate the probabilities of 8, 9, and 10 successes separately, and then add them together.
P(8 <= X <= 10) = P(X = 8) + P(X = 9) + P(X = 10)
Calculate each of these probabilities using the formula mentioned in part b).