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 calculate the probabilities in this scenario, we can use the concept of binomial probability. The formula for binomial probability is:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
where:
- P(X = k) refers to the probability of getting k successes (in this case, k teenagers with arachnophobia),
- nCk refers to the number of combinations of n items taken k at a time,
- p is the probability of success (in this case, the probability of a teenager having arachnophobia), and
- (1-p) is the probability of failure (in this case, the probability of a teenager not having arachnophobia).
(a) To calculate the probability that at least one of the teenagers suffers from arachnophobia, you can use the complement rule. The complement of "at least one" is "none." So, the probability that at least one teenager suffers from arachnophobia is equal to 1 minus the probability that none of them do.
Let's calculate it step by step:
Probability of a teenager having arachnophobia (p) = 7% = 0.07
Probability of a teenager not having arachnophobia (1-p) = 1 - 0.07 = 0.93
Number of teenagers in one tent (n) = 10
Probability that one specific teenager in a tent does not have arachnophobia = (1-p)^(n) = (0.93)^(10) = 0.528
Probability that one specific teenager in a tent has arachnophobia = 1 - 0.528 = 0.472
Probability that none of the teenagers in a tent have arachnophobia = (0.528)^10 = 0.034
Therefore, the probability that at least one teenager in a tent suffers from arachnophobia is 1 - 0.034 = 0.966, or 96.6%.
(b) To calculate the probability that exactly 2 of them suffer from arachnophobia, we use the binomial probability formula:
P(X = 2) = (10C2) * (0.07^2) * (0.93^8)
Using a calculator or software, you can determine that P(X = 2) ≈ 0.233, or 23.3%.
(c) To calculate the probability that at most 1 of them suffers from arachnophobia, we sum the probabilities of exactly 0 and exactly 1 person having arachnophobia:
P(X ≤ 1) = P(X = 0) + P(X = 1)
= [(10C0) * (0.07^0) * (0.93^10)] + [(10C1) * (0.07^1) * (0.93^9)]
Using a calculator or software, you can determine that P(X ≤ 1) ≈ 0.966 + 0.027, which is approximately 0.993, or 99.3%.
(d) If the camp counselor wants to make sure no more than 1 teenager in each tent is afraid of spiders, it seems reasonable for them to randomly assign teenagers to tents. Based on the probability calculations, the chance of having more than 1 teenager with arachnophobia in a tent is relatively low. Random assignment helps distribute the chances evenly and reduces the likelihood of any specific tent having multiple teenagers with arachnophobia.