Question

giving a test to a group of students, the grades and if they were a business majors are summarized below. one student is chosen at random. give your answer as a decimal out to at least 4 places.
a) find the probability that the student was a business major or got a grade of a.
b) find the probability that the student was a business major and got a grade of a.
c) find the probability that the student was a business major given they got a grade of a.
d) find the probability that the student did not get a b grade.
e) find ( p(bcup \text{business major}) ).
f) find ( p(c|\text{business major}) ).

Explanation:

Step1: Recall the formula for \(P(A\cup B)\)

The formula for \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)

Step2: Calculate \(P(\text{Business Major})\)

\(P(\text{Business Major})=\frac{39}{60}\)

Step3: Calculate \(P(\text{Grade A})\)

\(P(\text{Grade A})=\frac{23}{60}\)

Step4: Calculate \(P(\text{Business Major}\cap\text{Grade A})\)

\(P(\text{Business Major}\cap\text{Grade A})=\frac{16}{60}\)

Step5: Calculate \(P(\text{Business Major}\cup\text{Grade A})\)

\(P(\text{Business Major}\cup\text{Grade A})=\frac{39 + 23- 16}{60}=\frac{46}{60}\approx0.7667\)

Step6: Calculate \(P(\text{Business Major}\cap\text{Grade A})\)

\(P(\text{Business Major}\cap\text{Grade A})=\frac{16}{60}\approx0.2667\)

Step7: Recall the formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\)

Here \(A = \text{Business Major}\), \(B=\text{Grade A}\)
\(P(\text{Business Major}|\text{Grade A})=\frac{16}{23}\approx0.6957\)

Step8: Calculate \(P(\text{Not Grade B})\)

\(P(\text{Not Grade B})=1 - P(\text{Grade B})\)
\(P(\text{Grade B})=\frac{9}{60}\)
\(P(\text{Not Grade B})=1-\frac{9}{60}=\frac{51}{60}=0.85\)

Step9: Calculate \(P(B\cup\text{Business Major})\)

Using \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
\(P(B)=\frac{9}{60}\), \(P(\text{Business Major})=\frac{39}{60}\), \(P(B\cap\text{Business Major})=\frac{6}{60}\)
\(P(B\cup\text{Business Major})=\frac{9 + 39-6}{60}=\frac{42}{60}=0.7\)

Step10: Calculate \(P(C|\text{Business Major})\)

Using \(P(A|B)=\frac{P(A\cap B)}{P(B)}\)
\(P(C|\text{Business Major})=\frac{17}{39}\approx0.4359\)

Answer:

a) \(0.7667\)
b) \(0.2667\)
c) \(0.6957\)
d) \(0.85\)
e) \(0.7\)
f) \(0.4359\)