91Ó°ÊÓ

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.

Denote events

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

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

We are also given that

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

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.

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

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

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.