In a study of incentives used by companies to retain mature workers by The Conference Board, it was reported that 41% use flexible work arrangements. Suppose that of those companies that do not use flexible work arrangements, 30% give time off for volunteerism. In addition, suppose that of those companies that use flexible work arrangements, 50% give time off for volunteerism. If a company is randomly selected, determine the following probabilities:

a. The company uses flexible work arrangements or gives time off for volunteerism.
b. The company uses flexible work arrangements and does not give time off for volunteerism.
c. Given that the company does not give time off for volunteerism, the company uses flexible work arrangements.
d. The company does not use flexible work arrangements given that the company does give time off for volunteerism.
e. The company does not use flexible work arrangements or the company does not give time off for volunteerism.

Answer:

a) 0.587 = 58.7% probability that the company uses flexible work arrangements or gives time off for volunteerism.

b) 0.205 = 20.5% probability that the company uses flexible work arrangements and does not give time off for volunteerism.

c) 0.3317 = 33.17% probability that the company uses flexible work arrangements.

d) 0.4634 = 46.34% probability that the company does not use flexible work arrangements given that the company does give time off for volunteerism.

e) 0.795 = 79.5% probability that the company does not use flexible work arrangements or the company does not give time off for volunteerism.

Step-by-step explanation:

Conditional Probability

We use the conditional probability formula to solve this question. It is

[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]

In which

P(B|A) is the probability of event B happening, given that A happened.

[tex]P(A \cap B)[/tex] is the probability of both A and B happening.

P(A) is the probability of A happening.

a. The company uses flexible work arrangements or gives time off for volunteerism.

41% use flexible work arrangements.

Of the 100-41 = 59% that do not use flexible work arrangments, 30% give time off for volunteerism. So

[tex]p = 0.41 + 0.59*0.3 = 0.587[/tex]

0.587 = 58.7% probability that the company uses flexible work arrangements or gives time off for volunteerism.

b. The company uses flexible work arrangements and does not give time off for volunteerism.

41% use flexible work arrangements. Of those, 100 - 50 = 50% do not give time off for volunteerism. So

[tex]p = 0.41*0.5 = 0.205[/tex]

0.205 = 20.5% probability that the company uses flexible work arrangements and does not give time off for volunteerism.

c. Given that the company does not give time off for volunteerism, the company uses flexible work arrangements.

Here, we use conditional probability.

Event A: Does not give time off for volunteerism

Event B: Uses flexible work arrangments.

0.205 = 20.5% probability that the company uses flexible work arrangements and does not give time off for volunteerism.

This means that [tex]P(A \cap B) = 0.205[/tex]

Probability that the company does not give time off for volunteerism:

0.205(uses flexible work arrangments).

100 - 30 = 70% of 59%. So

[tex]P(A) = 0.205 + 0.7*0.59 = 0.618[/tex]

The desired probability is:

[tex]P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.205}{0.618} = 0.3317[/tex]

0.3317 = 33.17% probability that the company uses flexible work arrangements.

d. The company does not use flexible work arrangements given that the company does give time off for volunteerism.

Again conditional probability.

Event A: Gives time off for volunteerism

Event B: Does not use flexible work arrangements.

Probability of giving time off for volunteerism:

50% of 41%(have flexible work arrangements).

30% of 59%(don't have flexible work arrangements).

So

[tex]P(A) = 0.5*0.41 + 0.3*0.59 = 0.382[/tex]

Gives time off for volunteerism and does not use flexible work arrangements.

30% of 59%. So

[tex]P(A \cap B) = 0.3*0.59 = 0.177[/tex]

The desired probability is:

[tex]P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.177}{0.382} = 0.4634[/tex]

0.4634 = 46.34% probability that the company does not use flexible work arrangements given that the company does give time off for volunteerism.

e. The company does not use flexible work arrangements or the company does not give time off for volunteerism.

Does not use flexible work arrangments: 59%

Dont give time off for volunteerism(70% already in the 59%, so we dont add) and 50% of 41%.

[tex]p = 0.59 + 0.5*0.41 = 0.795[/tex]

0.795 = 79.5% probability that the company does not use flexible work arrangements or the company does not give time off for volunteerism.

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