Question

actually has allergy? test shows allergy? yes 90% yes 8% no 10% no 92% yes 15% no 85% click the arrows to choose an answer from each menu. the probability that a child will receive a positive test is choose... . for a child with a positive test, the probability that the child actually has a food allergy is approximately choose... . the new test choose... likely to be a reliable indicator that the child has a food allergy.

Explanation:

Step1: Calculate the probability of a positive test

Let $P(A)$ be the probability of actually having an allergy, $P(
eg A)$ be the probability of not having an allergy, $P(+|A)$ be the probability of a positive - test given having an allergy, and $P(+|
eg A)$ be the probability of a positive - test given not having an allergy.
$P(A)=0.08$, $P(
eg A)=0.92$, $P(+|A) = 0.9$, $P(+|
eg A)=0.15$.
Using the law of total probability, $P(+)=P(+|A)P(A)+P(+|
eg A)P(
eg A)$.
$P(+)=0.9\times0.08 + 0.15\times0.92$
$P(+)=0.072+0.138$
$P(+)=0.21$

Step2: Calculate the probability of having an allergy given a positive test

Using Bayes' theorem, $P(A|+)=\frac{P(+|A)P(A)}{P(+)}$.
We know $P(+|A)P(A)=0.9\times0.08 = 0.072$ and $P(+)=0.21$.
So $P(A|+)=\frac{0.072}{0.21}\approx0.343$

Answer:

The probability that a child will receive a positive test is $0.21$. For a child with a positive test, the probability that the child actually has a food allergy is approximately $0.343$. The new test is not very likely to be a reliable indicator that the child has a food allergy.