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Worked solution

Give proper fractions and condition on both people finding jobs

Express all answers as proper fractions. In 2013,60% of the unemployed citizens of country A were graduates. Within a year,70% of these unemployed graduates found a job, while 40% of the unemployed non-graduates found one.

(i) A person unemployed in 2013 is chosen at random. Find the probability the person is not a graduate, given no job has been found. marks

(ii) Two people unemployed in 2013 are chosen at random. Find the probability that

(a) exactly one found a job within a year, marks

(b) exactly one is a graduate, given both found jobs within a year. marks

[2014/ACJC/II/10]

CJC · Review problems · Question 13

Answer

(i) 4/74/7; (ii)(a) 609/1250609/1250; (b) 336/841336/841.

(i) Read the full source tree

The graduate/non-graduate branches are 0.6/0.4. Job/no-job rates are 0.7/0.3 for graduates and 0.4/0.6 for non-graduates. Hence P(non-graduate∣no job)=(0.4)(0.6)/[(0.6)(0.3)+(0.4)(0.6)]=4/7P(\text{non-graduate}\mid\text{no job})=(0.4)(0.6)/[(0.6)(0.3)+(0.4)(0.6)]=4/7.

How did this part go?

(ii)(a) Add the two ordered possibilities

P(no job)=(0.6)(0.3)+(0.4)(0.6)=21/50P(\text{no job})=(0.6)(0.3)+(0.4)(0.6)=21/50, so P(job)=29/50P(\text{job})=29/50. Under the source’s independent-person selection model, the probability of exactly one employed person is (29/50)(21/50)+(21/50)(29/50)=609/1250(29/50)(21/50)+(21/50)(29/50)=609/1250.

How did this part go?

(ii)(b) Joint numerator and both-employed denominator

The favourable ordered pair is an employed graduate and an employed non-graduate, in either order. Thus

P(exactly one graduate∣both employed)=(0.6)(0.7)(0.4)(0.4) 2!(29/50)(29/50)=336/841.\begin{aligned}&P(\text{exactly one graduate}\mid\text{both employed})\\ & =\frac{(0.6)(0.7)(0.4)(0.4)\,2!}{(29/50)(29/50)}\\ & =336/841.\end{aligned}

All final answers are proper fractions as requested. These two-person products use the source’s independence model; a specified finite population sampled without replacement would need its counts for an exact correction.

How did this part go?

Compare your sketch

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Graduate probability 0.6, Non-graduate probability 0.4. Given Graduate: Job 0.7, No job 0.3. Given Non-graduate: Job 0.4, No job 0.6. All source branches retained.
Graduate probability 0.6, Non-graduate probability 0.4. Given Graduate: Job 0.7, No job 0.3. Given Non-graduate: Job 0.4, No job 0.6. All source branches retained.
Graduate probability 0.6, Non-graduate probability 0.4. Given Graduate: Job 0.7, No job 0.3. Given Non-graduate: Job 0.4, No job 0.6. All source branches retained.

Common mistakes

Do not use the original 60% graduate rate inside the employed-only population without conditioning.

Solution notes

Source proper-fraction instruction, all three asks, complete tree, exact intermediate marginals, ordered-pair products and marks retained. Independent-person model underlying source products is explicit.

Read the hint