First graders who misbehave in school may be more likely to be regular smokers as young adults according to a new study presented in the July 2004 issue of the American Journal of Epidemiology. After following a group of U.S. first graders for 15 years, it was found that among those kids who had tried smoking and misbehaved, 72% were daily smokers. (round to 3 decimal places)
(b) What is the probability that exactly 8 of the next 12 randomly selected young adults who misbehaved in early grades and have tried smoking are daily smokers?
(c) What is the probability that exactly 20 of the next 30 randomly selected young adults who misbehaved in early grades and have tried smoking are daily smokers?
To solve these probability problems, we need to use the binomial probability formula.
The binomial probability formula calculates the probability of having a specific number of successes in a fixed number of trials, given a specific probability of success in each trial.
The formula for the probability of exactly 'r' successes in 'n' trials is:
P(X = r) = nCr * p^r * (1-p)^(n-r)
where:
- nCr represents the number of combinations of 'n' items taken 'r' at a time.
- p is the probability of success in a single trial.
- q = 1-p represents the probability of failure in a single trial.
- X is a random variable representing the number of successes.
Now let's calculate the probabilities:
(b) What is the probability that exactly 8 of the next 12 randomly selected young adults who misbehaved in early grades and have tried smoking are daily smokers?
From the given information, we know that 72% of kids who had tried smoking and misbehaved were daily smokers. This means that the probability of success (p) is 0.72 and the probability of failure (q) is 1 - p = 1 - 0.72 = 0.28.
Using the binomial probability formula, we can calculate:
P(X = 8) = 12C8 * (0.72)^8 * (0.28)^(12-8)
To calculate 12C8, we can use the formula:
12C8 = 12! / (8!(12-8)!)
Thus, the probability of exactly 8 out of 12 young adults being daily smokers would be:
P(X = 8) = (12! / (8!(12-8)!)) * (0.72)^8 * (0.28)^(12-8)
(c) What is the probability that exactly 20 of the next 30 randomly selected young adults who misbehaved in early grades and have tried smoking are daily smokers?
Applying the same logic as in part (b), we can calculate the probability of exactly 20 out of 30 young adults being daily smokers:
P(X = 20) = 30C20 * (0.72)^20 * (0.28)^(30-20)
To calculate 30C20, we can use the formula:
30C20 = 30! / (20!(30-20)!)
Now, using the binomial probability formula, we can calculate the probability.
Keep in mind that these calculations involve factorials and can be time-consuming. You may want to use a calculator or a statistical software tool to get the exact values.