Question

in this problem, round to 4 decimals. food allergies affect an estimated 18% of children under age 5 in the us. what is the probability that in a kindergarten class of 19 kids under age 5 at least one of them has food allergies?

Explanation:

Step1: Find the probability of a child not having food allergies

The probability that a child has food allergies is $p = 0.18$. So the probability that a child does not have food allergies is $q=1 - p=1 - 0.18 = 0.82$.

Step2: Find the probability that none of the 19 kids have food allergies

Since the events of each child not having food allergies are independent, the probability that none of the $n = 19$ kids have food allergies is $q^{n}=0.82^{19}$.
$0.82^{19}\approx0.0147$.

Step3: Find the probability that at least one kid has food allergies

The probability that at least one kid has food allergies is the complement of the event that none of the kids have food allergies. Let $P(X\geq1)$ be the probability that at least one kid has food allergies. Then $P(X\geq1)=1 - P(X = 0)$.
$P(X\geq1)=1-0.82^{19}\approx1 - 0.0147=0.9853$.

Answer:

$0.9853$