Question

According to July \(31, 2013\), posting on cnn.com subsequent to the death of a child who bit into a peanut, a \(2010\) study in the journal Pediatrics found that \(8\% \)of children younger than eighteen in the United States have at least one food allergy. Among those with food allergies, about \(39\% \)had a history of severe reaction.

a. If a child younger than eighteen is randomly selected, what is the probability that he or she has at least one food allergy and a history of severe reaction?

b. It was also reported that \(30\% \) of those with an allergy in fact are allergic to multiple foods. If a child younger than eighteen is randomly selected, what is the probability that he or she is allergic to multiple foods?

Short answer

a. The probability that he or she has at least one food allergy and a history of severe reaction is \(0.0312\).

b. The probability that he or she is allergic to multiple foods is \(0.024\).

Step-by-step solution

Definition of Probability

Probability is an area of mathematics that deals with numerical descriptions of the likelihood of an event occurring or a proposition being true. An event's probability is a number between \(0{\rm{ }}and{\rm{ }}1\), with zero indicating impossibility and one indicating certainty, broadly speaking.

Given Data for part a.

Denote events

\(A = \{ child\;has\;a\;food\;allergy\} ;\)

\(B = \{ child\;has\;a\;history\;of\;severe\;reaction\} \);

We are also given that

\(\begin{array}{l}P(A) &= &0.08\\P(B\mid A) &=& 0.39\end{array}\)

Calculation for the determination of probability in part a.

The Multiplication Rule

\(P(A \cap B) = P(A\mid B) \cdot P(B)\)

Using the multiplication rule, we have

\(\begin{array}{c}P(B \cap A) &=& P(B\mid A) \cdot P(A)\\ &=& 0.39 \cdot 0.08\\ &=& 0.0312.\end{array}\)

This is the probability that the child has at least one food allergy and a history of a severe reaction.

Given data for part b.

Denote events

\(\begin{array}{l}A = \{ {\rm{ child has a food allergy }}\} ;\\B = \{ {\rm{ child has a history of severe reaction }}\} ;\end{array}\)

We are also given that

\(\begin{array}{l}P(A) &=& 0.08\\P(B\mid A) &=& 0.39\end{array}\)

Calculation for the determination for part b.

The Multiplication Rule

\(P(A \cap B) = P(A\mid B) \cdot P(B)\)

Denote event

\(C = \{ a\;child\;younger\;than\;18\;is\;allergic\;to\;multiple\;foods\} \).

We are given that

\(P(C\mid A) = 0.3.\;\;\;(30\% )\)

Again, using the multiplication rule we have

\(P(C)\mathop = \limits^{(1)} P(C \cap A)\mathop = \limits^{(2)} P(A) \cdot P(C\mid A) = 0.08 \cdot 0.3 = 0.024\)

(1): here we use the fact that \(C \subset A\), which indicates that \(C = C \cap A\). This is true because if concept that a person has a food allergy is bigger concept than the person has multiple allergies,

(2): multiplication rule.