Question

A 2005 survey found that 7% of teenagers (ages 13 to 17) suffer from an extreme fear of spiders (arachnophobia). At a summer camp there are 10 teenagers sleeping in each tent. Assume that these 10 teenagers are independent of each other. What is the probability that at least one of them suffers from arachnophobia

Answer 1

Probability that at least one of them suffers from arachnophobia is 0.5160.

Explanation:

We are given that a 2005 survey found that 7% of teenagers (ages 13 to 17) suffer from an extreme fear of spiders (arachnophobia).

Also, At a summer camp there are 10 teenagers sleeping in each tent.

Firstly, the binomial probability is given by;

`P(X=r) =\binom{n}{r}p^(r)(1-p)^(n-r) for x = 0,1,2,3,....`

where, n = number of trials(teenagers) taken = 10

r = number of successes = at least one

p = probability of success and success in our question is % of

the teenagers suffering from arachnophobia, i.e. 7%.

Let X = Number of teenagers suffering from arachnophobia

So, X ~
`Binom(n= 10,p=0.07)`

So, probability that at least one of them suffers from arachnophobia

= P(X >= 1) = 1 - probability that none of them suffers from arachnophobia

= 1 - P(X = 0) = 1 -
`\binom{10}{0}0.07^(0)(1-0.07)^(10-0)`

= 1 - (1 * 1 *
`0.93^(10)` ) = 1 - 0.484 = 0.5160 .

Therefore, Probability that at least one of them suffers from arachnophobia is 0.5160 .