# Problem 411

The table shows the result of a survey in  which 143 men and 145 women workers ages 25 to 64 were asked if they have at least one month’s set aside for emergencies.

 Men Women Total Less than one month’s income 64 85 149 One month’s income or more 79 60 139 Total 143 145 288

The empirical probability of an event E is the relative frequency of event E.

P(E) =

(a)    Find the probability that a randomly selected worker has one month’s income or more set aside for emergencies.

The total number of surveyed workers id 288, and 139 of than have one month’s income or more set aside for emergencies

Using the empirical probability formula, find the probability P(K), where K is the events of having one month’s income or more set aside for emergencies.

P(K) =

0.483

(b)   Given that a randomly selected worker is a male, find the probability that the worker has less than one month’s income.

A conditional probability is the probability of the event occurring, given that another event has .

The number of males among the asked workers is 143, and 64 of them have less than one month’s income.

Let L be the event of having less than one month’s income and let M be the event that a worker is male.

Using the empirical probability formula, find P(L│M), the probability that a randomly selected worker have less than one month’s income given that the worker is a male.

P(L│M) =

0.448

(c)    Given that a randomly selected worker has one month’s income or more, find the probability that the worker is a female.

Form part(a), there are 139 workers with one month’s income or more. There are 60 females among those 139 workers.

Let F be the events that a worker is female.

Using the empirical probability formula, find P(F│K), the probability that a randomly selected worker is a female given that the worker has one month’s income or more.

P(F│K) =

0.432

(d)   Are the events “having less than one month’s income saved” and “being male” independent?

Two events are independent if the occurrence of one if the events does not affect the probability of the occurrence of the other event. The condition of independence of two events L and M is the following.

P(L│M) = P(L)

First find P(L). 149 of 288 workers have less than one month’s income.

Using the empirical probability formula, find P(L), the probability of having less than one month’s income saved.

P(L) =

0.517

Recall form part (b) that P(L│M) = 0.448. AS P(L│M) ≠ P(L), the events “having less than one month’s income saved” and “being male” are dependent.