A preliminary survey shows that 35% of college students smoke. In a class of 42 students, what is the probability that more than half the students smoke?

To calculate the probability that more than half of the students in a class of 42 smoke, we need to determine the number of students that constitute more than half of the class.

Step 1: Calculate the number of students that make up more than half the class.
Half of the class would be (42 / 2) = 21 students.
To find the number of students that make up more than half the class, subtract 21 from the total number of students in the class: 42 - 21 = 21 students.

Step 2: Calculate the probability that a college student smokes.
According to the preliminary survey, 35% of college students smoke. So, the probability that a college student smokes is 0.35.

Step 3: Use the binomial probability formula to find the probability that more than half of the students smoke.
The binomial probability formula is: P(X > k) = 1 - P(X <= k),
where X = number of successes (students who smoke),
k = number of students that make up more than half of the class.

In this case, X follows a binomial distribution with parameters n = 42 and p = 0.35 (the probability of success).

P(X > k) = 1 - P(X <= k)
P(X > 21) = 1 - P(X <= 21)

Step 4: Calculate P(X <= 21).
To find P(X <= 21), we need to calculate the cumulative probability up to 21, using the binomial probability distribution.

P(X <= 21) = ∑[k=0 to 21] (42 choose k) * 0.35^k * (1-0.35)^(42-k)

Step 5: Calculate P(X > 21).
Using the formula P(X > k) = 1 - P(X <= k), we can find the probability that more than half of the students smoke.

P(X > 21) = 1 - P(X <= 21)

By following these steps and performing the calculations, you can find the probability that more than half of the students in the class smoke.