In a study of incentives used by companies to retain mature

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, 20% 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.

Solution

Let\'s just start off with what we know. The probability that a company uses flexible work arrangements is 41% Given that a company does not use flexible work arrangements, the probability that they use time off for volunteerism is 30% Given that the company does use flexible work arrangements, the probability that they use time off for volunteerism is 60% A) The company uses flexible work arrangements or gives time off for volunteerism. So we know that 41% of companies use flexible work arrangements. That means that 59% do not. Of that 59%, 30% give time off for volunteerism. That means that (.59)*(.30) = 0.177 17.7% of companies do not use flexible work arrangements but do give time off for volunteerism. Now we can just add the probability of a company using flexible work arrangements (0.41) to the probability of the company not using flexible work schedules, but giving time off (0.177) 0.41+0.177 = 0.587 or 58.7% B) The company uses flexible work arrangements and does not give time off for volunteerism. The probability that a company uses flexible work arrangements is 41% Given that the company does use flexible work arrangements, the probability that they use time off for volunteerism is 60% That means that, given that the company does use flexible work arrangements, the probability that they do not use time off for volunteerism is 40% So we need to multiply the probability that the company uses flexible work schedules (0.41) by the probability that that company does not use time off for volunteerism (0.4) (0.41)*(0.4) = 0.164 0r 16.4% C) Given that the company does not give time off for volunteerism, the company uses flexible work arrangements. This one is a little tricky, because you have to use Bayes\' Theorem First, we need to calculate the probability that a company does not give time off... (0.41)*(0.40) + (0.59)*(0.60) = 0.518 We know the probability that the company uses flexible work arrangements is 0.41 The last thing we need to know is the probability that a company that uses flexible work arrangements does not give time off which is 0.40 Now the equation is: P(A|B) = (0.40*0.41)/(0.518) = 0.3166 or 31.66% D) The company does not use flexible work arrangements given that the company does give time off for volunteerism. We know that given that a company does not use flexible work arrangements, the probability that they use time off for volunteerism is 30% So we need to flip that around using Bayes\' Theorem again. That probability that a company does not use flexible work schedules is 0.59 From what we have done already, we know that the probability that a company uses time off is 1-0.518 = 0.482 (0.30*0.59) / (0.482) = 0.3672 or 36.72% E) The company does not use flexible work arrangements or the company does not give time off for volunteerism. This is going to be the sum of the probability that a company uses flexible work schedules ( 0.41) and the probability that a company does not give off for volunteerism and that they do not use flexible work schedules. We know that given that a company does not use flexible work arrangements, the probability that they use time off for volunteerism is 30% That means that given that a company does not use flexible work arrangements, the probability that they do not use time off for volunteerism is 70% 59% do not use flexible work schedules and 70% of those do not give time off for volunteerism. So (.59)*(.70) = 0.413 or 41.3% of companies do not use flexible work schedules and do not give time off for volunteerism. So the probabilities we need to add are (.41) and (0.413). That gives us 0.823 or 82.3% DONE!