Question

Experience shows that 20% of all participants in the Statistics 1 exam achieve a good grade. 40% of participants are female and 18% of female participants write a good exam.

a) What is the probability that a randomly selected student is a man who has not written a good exam?

b) A student has written a good exam. What is the probability that it was a woman?

c) The probability in statistic 2 to write a good exam is 0.15. Calculate the probability that a student is good in both exams if the results of both exams are independent of each other.

d) A student who was already good in statistic 1 is also good in statistic 2 with a probability of 0.3. Under these new conditions, calculate the probability that a student is good in both exams.

e ) A randomly selected student has written a good Statistics 2 exam.
Under the same conditions as in d), what is the probability that his statistic 1 exam was also good, if a student with a not good statistic 1 exam writes a good statistic 2 exam in 20% of all cases?

What is the probability that a student with a not good statistic 1 exam would have to write a good statistic 2 exam in order for the results of the two exams to be independent of each other?

What would be the probability if it were known that the events "good statistic 1 exam" and "good statistic 2 exam" are disjoint and that the overall probability calculated in e) to write a good statistic 2 exam should not change.

Answer 1

(there are more than 4 parts, as per policy i am answering first 4 parts)

a.

P(good exam) = P(man, good exam) + P(woman, good exam)

= P(man, good exam) + P(woman)*P(good exam | woman)

0.20 = P(man, good exam) + 0.40*0.18

P(man, good exam) = 0.20 - 0.40*0.18 = 0.128

P(man, not good exam) = P(man) - P(man, good exam)

= (1-0.40) - 0.128

= 0.472

b.

P(woman | good exam) = P(woman)*P(good exam | woman) / P(good exam)

= 0.40*0.18 / (0.20)

= 0.36

c.

given they are independent, therefore :

P(both good) = P(1 good)*P(2 good)

= 0.2*0.15

= 0.03

d.

P(both good) = P(1 good)*P(2 good | 1 good)

= 0.20*0.30

= 0.06

(please upvote)