Question

3.32 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 12 teenagers sleeping in each tent. assume that these 12 teenagers are independent of each other. (in parts a, b, c, do not enter your answer after multiplying by 100. for example, if your answer is 0.4266, then enter it as 0.4266, not as 42.66.) (a) calculate the probability that at least one of them suffers from arachnophobia. (please round to four decimal places) (b) calculate the probability that exactly 2 of them suffer from arachnophobia? (please round to four decimal places) (c) calculate the probability that at most 1 of them suffers from arachnophobia? (please round to four decimal places)

Explanation:

Step1: Identify probabilities and number of trials

The probability of a teenager having arachnophobia $p = 0.07$, the probability of not having arachnophobia $q=1 - p= 0.93$, and the number of teenagers $n = 12$.

Step2: Calculate probability for part (a)

The probability that none of them has arachnophobia is $P(X = 0)=C(n,0)\times p^{0}\times q^{n}$, where $C(n,k)=\frac{n!}{k!(n - k)!}$. So $P(X = 0)=\binom{12}{0}\times(0.07)^{0}\times(0.93)^{12}=1\times1\times(0.93)^{12}\approx0.4186$. The probability that at least one has arachnophobia is $P(X\geq1)=1 - P(X = 0)=1-(0.93)^{12}\approx1 - 0.4186 = 0.5814$.

Step3: Calculate probability for part (b)

The binomial - probability formula is $P(X = k)=C(n,k)\times p^{k}\times q^{n - k}$. For $k = 2$, $n = 12$, $p = 0.07$ and $q = 0.93$, we have $P(X = 2)=\binom{12}{2}\times(0.07)^{2}\times(0.93)^{10}=\frac{12!}{2!(12 - 2)!}\times(0.07)^{2}\times(0.93)^{10}=66\times0.0049\times0.483982\approx0.1586$.

Step4: Calculate probability for part (c)

$P(X\leq1)=P(X = 0)+P(X = 1)$. $P(X = 0)=(0.93)^{12}\approx0.4186$, and $P(X = 1)=\binom{12}{1}\times(0.07)^{1}\times(0.93)^{11}=12\times0.07\times0.483982\approx0.4085$. So $P(X\leq1)=0.4186 + 0.4085=0.8271$.

Answer:

(a) 0.5814
(b) 0.1586
(c) 0.8271