Arachnophobia. A 2005 Gallup Poll found that 7% 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.52
(a) Calculate the probability that at least one of them suffers from arachnophobia.
(b) Calculate the probability that exactly 2 of them suffer from arachnophobia.
(c) Calculate the probability that at most 1 of them suffers from arachnophobia.
(d) If the camp counselor wants to make sure no more than 1 teenager in each tent is afraid of
spiders, does it seem reasonable for him to randomly assign teenagers to tents?
To solve this problem, we need to use the concept of probability. Probability is a measure of the likelihood of an event occurring. In this case, we're looking at the probability of a teenager suffering from arachnophobia in a tent.
(a) To calculate the probability that at least one teenager suffers from arachnophobia, we calculate the probability that none of them suffer from arachnophobia and subtract it from 1.
The probability of one teenager not suffering from arachnophobia is 1 - 0.07 = 0.93. As there are 10 teenagers in each tent, the probability that none of them suffer from arachnophobia is (0.93)^10, since their fears are independent.
So the probability that at least one teenager in a tent suffers from arachnophobia is 1 - (0.93)^10.
(b) To calculate the probability that exactly 2 of them suffer from arachnophobia, we use the binomial probability formula:
P(X=k) = (n C k) * (p^k) * ((1-p)^(n-k))
where n is the number of trials (number of teenagers in the tent), k is the number of successful outcomes (2 teenagers with arachnophobia), p is the probability of success (0.07), and (n C k) represents the binomial coefficient.
The binomial coefficient (n C k) is calculated as n! / (k! * (n-k)!)
So the probability is (10 C 2) * (0.07^2) * ((1-0.07)^(10-2)).
(c) To calculate the probability that at most 1 of them suffers from arachnophobia, we need to find the probability of 0 or 1 teenager suffering from arachnophobia.
The probability of 0 teenagers suffering from arachnophobia is calculated as (0.93)^10.
The probability of 1 teenager suffering from arachnophobia is calculated as 10 * (0.07) * (0.93^9), since there are 10 ways to choose which teenager suffers from arachnophobia.
So the probability of at most 1 teenager suffering from arachnophobia is (0.93)^10 + 10 * (0.07) * (0.93^9).
(d) To determine if it's reasonable to randomly assign teenagers to tents, we need to consider the probability of having more than 1 teenager with arachnophobia in a tent.
The probability of having more than 1 teenager with arachnophobia in a tent is 1 - probability of at most 1 teenager suffering from arachnophobia calculated in part (c).
If this probability is sufficiently low, the camp counselor could reasonably assume that randomly assigning teenagers to tents will ensure that no more than 1 teenager in each tent is afraid of spiders. The specific threshold for "sufficiently low" would depend on the counselor's risk tolerance and the camp's policy.
Remember to substitute the appropriate values in the calculations based on the given information.