Employment data at a large company reveal that 58 % of the workers are married, that 41 % are college graduates, and that 1/2 of the college graduates are married. What is the probability that...
Employment data at a large company reveal that 58 % of the
workers are married, that 41 % are college graduates, and
that 1/2 of the college graduates are married.
What is the probability that a randomly chosen worker is:
(b) Married but not a college graduate?
(c) Married or a college graduate?
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1 Answer
We are given that 58% of the workers are married and that 41% are graduates. Also, 1/2 of the graduates are married.
I. We can set up a probability tree:
Let G be the probability that a worker is a graduate, so G' is the probability that she is not a graduate. Let M be the probability that he is married and M' the probability of being single.
.41 G  .5 M .205

.5 M' .205
.59 G'  M

 M'
Since the percentage of married workers is 58%, the sum of married graduates and married nongraduates is .58. Thus .41(.5)+.59x=.58 where x is the probability of being a married nongraduate. `x~~.6356 ` Then the probabilities in the right column of the tree sum to 1 (each employee is either a graduate or not, etc... So we have:
.41 G  .5 M .205

 .5 M' .205
.59 G'  .6356 M .375

 .3644 M' .215
(a) The probability of being married but not a graduate is (.59)(.6356)=.375
(b) The probability of being married or a graduate is 1 minus the complement or 1(probability of being a nonmarried nongraduate) or 1.215=.785
II. An alternative is to draw a Venn diagram  two intersecting circles representing graduates and married employees.
For the graduates the circle total is .41; 1/2 are in the intersection so label the intersection .205 and G not in the intersection as .205. Since married is .58, and .205 is in the intersection, we have .58.205=.375 in married but not in the intersection. This leaves .215 in the space outside either circle.
III. Use Bayesian probabilities:
`P(AT)=(P(TA)P(A))/(P(TA)P(A)+P(TA')(P(A'))) `
For (a) P(MG')=`(P(G'M)P(M))/(P(G'M)P(M)+P(G'M')P(M')) `