Question

A poll found that 8% of teenagers (ages 13 to 17) suffer from arachnophobia and are extremely afraid of spiders. At a summer camp there are 10 teenagers sleeping in each tent. Assume that these 10 teenagers are independent of each other. (Round your answers to four decimal places.) (a) Calculate the probability that at least one of them suffers from arachnophobia.

Answer 1

To calculate the probability that at least one of the 10 teenagers in a tent suffers from arachnophobia, you can use the complement rule. First, find the probability that none of them suffer from arachnophobia and then subtract that probability from 1 (since the sum of all possible outcomes should equal 1).

The probability that one teenager does not suffer from arachnophobia is 1 - 0.08 (because 8% of teenagers suffer from arachnophobia), which is 0.92.

Since the teenagers in the tent are independent of each other, you can calculate the probability that all of them do not suffer from arachnophobia by multiplying the probabilities together:

P(None suffer from arachnophobia) = (0.92)^10

Now, calculate this probability:

P(None suffer from arachnophobia) ≈ 0.5132 (rounded to four decimal places)

Now, use the complement rule to find the probability that at least one of them suffers from arachnophobia:

P(At least one suffers from arachnophobia) = 1 - P(None suffer from arachnophobia)

P(At least one suffers from arachnophobia) ≈ 1 - 0.5132 ≈ 0.4868 (rounded to four decimal places)

So, the probability that at least one of the 10 teenagers in the tent suffers from arachnophobia is approximately 0.4868, or 48.68%.