Question

a high school science teacher has 78 students. of those students, 35 are in the band and 32 are on a sports team. there are 16 students who are not in the band or on a sports team. one student from the 78 students will be selected at random. let event b represent the event of selecting a student in the band, and let event s represent the event of selecting a student on a sports team. are b and s mutually exclusive events? a no, because ( p(bcap s)=\frac{5}{78} ). b no, because ( p(bcap s)=\frac{48}{78} ). c no, because ( p(bcap s)=\frac{62}{78} ). d yes, because ( p(bcap s)=\frac{5}{78} ). e yes, because ( p(bcap s)=\frac{62}{78} ).

Explanation:

Step1: Calculate the number of students in \(B\cup S\)

The total number of students is \(n = 78\). The number of students not in \(B\cup S\) is \(16\). So, \(n(B\cup S)=78 - 16=62\).

Step2: Use the formula \(n(B\cup S)=n(B)+n(S)-n(B\cap S)\)

We know \(n(B) = 35\), \(n(S)=32\), and \(n(B\cup S)=62\). Substitute into the formula: \(62=35 + 32-n(B\cap S)\).

Step3: Solve for \(n(B\cap S)\)

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Step4: Calculate \(P(B\cap S)\)

The probability \(P(B\cap S)=\frac{n(B\cap S)}{n}=\frac{5}{78}\).

Step5: Determine if \(B\) and \(S\) are mutually - exclusive

Two events \(B\) and \(S\) are mutually exclusive if \(P(B\cap S)=0\). Since \(P(B\cap S)=\frac{5}{78}
eq0\), \(B\) and \(S\) are not mutually exclusive.

Answer:

A. No, because \(P(B\cap S)=\frac{5}{78}\)