Question

a company has both male and female employees. the company has shirts and jackets with the company logo to give away to employees. for each of the company’s 209 employees, a manager asked which piece of clothing the employee prefers. the preferences, based on gender, are summarized in the table below.

	male	female
jacket	93	46

suppose an employee of the company is chosen at random.
answer each part. do not round intermediate computations, and round your answers to the nearest hundredth.
(if necessary, consult a list of formulas.)
(a) what is the probability that the employee prefers a shirt?
(b) what is the probability that the employee is male or prefers a shirt?

Explanation:

Part (a)

Step 1: Find total employees

Total employees = \(36 + 34 + 93 + 46 = 209\)

Step 2: Find number of employees who prefer a shirt

Number of employees who prefer a shirt = \(36 + 34 = 70\)

Step 3: Calculate probability

Probability = \(\frac{\text{Number of employees who prefer a shirt}}{\text{Total number of employees}}=\frac{70}{209}\approx0.33\)

Part (b)

Step 1: Find number of male employees

Number of male employees = \(36 + 93 = 129\)

Step 2: Find number of employees who prefer a shirt (already found as 70)

Step 3: Find number of male employees who prefer a shirt (36, from the table)

Step 4: Apply the formula for \(P(A \cup B)=P(A)+P(B)-P(A \cap B)\)

Let \(A\) be the event that the employee is male and \(B\) be the event that the employee prefers a shirt.
\(P(A)=\frac{129}{209}\), \(P(B)=\frac{70}{209}\), \(P(A \cap B)=\frac{36}{209}\)
\(P(A \cup B)=\frac{129 + 70 - 36}{209}=\frac{163}{209}\approx0.78\)

Answer:

(a) \(\approx 0.33\)
(b) \(\approx 0.78\)