Probability question I'm having difficulty with

To gauge the relationship between education and unemployment an economist turned to the US Census, from which the following table was produced:

(First number represents employed, second number represents unemployed)

Not a high school graduate 0.0975 , 0.0080

High school graduate 0.3108 , 0.0128

Some college, no degree 0.1785 , 0.0062

Associate’s degree 0.0849 , 0.0023

Bachelor’s degree 0.1959 , 0.0041

Advanced degree 0.0975 , 0.0015

a. What is the probability that a high school graduate is unemployed?

b. Determine the probability that a randomly selected individual is employed.

c. Find the probability that an unemployed person possesses an advanced degree.

d. What is the probability that a randomly selected person did not finish high school?

Data from the Office on Smoking and Health, Centers for Disease Control and Prevention, indicate that 40% of adults who did not finish high school, 34% of high school graduates, 24% of adults who completed some college (no degree), and 14% of graduates (Degree holders) smoke.

e. Suppose that one individual is selected at random and it is discovered that the individual smokes. What is the probability that the individual is a graduate? (Use the probabilities from the table above to solve this question)

HINT:

Conditional probability: P(A|B) = P(A and B)/P(B)

Bayes’ Law Formula:

P(Ai |B)= ]P(Ai)P(B|Ai)] / [P(A1 )P(B | A1 ) + P(A 2 )P(B | A 2 ) + . . . + P(A k )P( B | A k )]

Re: Probability question I'm having difficulty with

Have you tried anything at all?

Personally, I would not use "Bayes formula" or the "Conditional probability" formula.

Instead I would calculate specific numbers. Suppose our pool consists of 1000000 people. Then

"Not a high school graduate 0.0975 , 0.0080"

So we have 0.0975*1000000= 97500 non-high school graduates who are employed, 0.0080*1000000=8000 who are unemployed.

"High school graduate 0.3108 , 0.0128"

So we have 0.3108*1000000= 310800 high school graduates who are employed, 0.0128*1000000= 12800 high school graduates who are unemployed.

"Some college, no degree 0.1785 , 0.0062"

So we have .01785*1000000= 178500 people with some college who are employed, 0.0062*1000000= 6200 people with some college who are unemployed.

"Associate’s degree 0.0849 , 0.0023"

So we have 0.0849*1000000= 84900 people with associate's degree who are employed, 0.0023*1000000= 2300 people with associate's degree who are unemployed.

"Bachelor’s degree 0.1959 , 0.0041"

So we have 0.1959*1000000= 195900 people with bachelor's degree who are employed, 0.0041*1000000= 4100 people with bachelor's degree who are unemployed.

"Advanced degree 0.0975 , 0.0015"

So we have 0.0975*1000000= 97500 people with advanced degree who are employed, 0.0015*1000000= 1500 people with advanced degree who are unemployed.

Now, the first question was "What is the probability that a high school graduate is unemployed?". Okay, there are 310800 high school graduates who are employed, 12800 high school graduates who are not employed. There are a total of 310800+ 12800= 323600 high school graduates, 12800 of whom are not employed. The probability a high school graduate is unemployed is [tex]\frac{12800}{323600}= 0.040, approximately.

The second question was "Determine the probability that a randomly selected individual is employed."

Add up the numbers of employed in each group and divide by 1000000, the total number of people.

Third was "The probability that an unemployed person has an advanced degree". There were 1500 people with advanced degrees who were unemployed. Find the total number of unemployed persons and divide 1500 by that.

Fourth was "The probability that a randomly selected person did not finish high school". There were 97500 non high school graduates who were employed, 8000 who were not so a total of 97500+ 8000= 105500 non high school graduates out of 1000000. The probability of any one of those 1000000 people being a non high school graduate is $\displaystyle \frac{105500}{1000000}$.

(e) gives further information: "34% of high school graduates, 24% of adults who completed some college (no degree), and 14% of graduates (Degree holders) smoke."

Since there were 310800+ 12800= 323600 and 34% of them smoke, there are .34(323600)= 110024 high school graduates who smoke. We can, similarly, calculate the number of adults who completed some college who smoke and the number of graduates who smoke. With that information, "the probability that the individual who smokes is a graduate" is the number of graduates who smoke divided by the total number of smokers.