Question

In some states, high school students must pass a graduation test before they are able to receive their high school diplomas. Suppose the students are given up to three chances to pass the exam. If they pass it, they don't need to take it again. If they do not pass the test on the first try, they take it a second time. If they don't pass on the second try, they take it a third time. From past data, we know that 78% of students pass the test the first time, while 85% of students who take it a second time pass on the second try, and 60% of students who take it a third time pass on the third try.

a) What percent of students are not allowed to graduate because of their performance on the exam? Enter your answer as a percent, rounded to the nearest tenth of a percent

b) What percent of students take the test three times? Enter your answer as a percent, rounded to the nearest tenth of a percent

c) What percent of students pass a graduation test and receive their high school diplomas? Enter your answer as a percent, rounded to the nearest tenth of a percent.

Answer 1

P(pass in 1st time) = 0.78

P(pass in 2nd time) = 0.85

P(pass in 3rd time) = 0.60

(a) P(not graduate) = P(fail all three times)

= (1 - P(pass in 1st time))*(1 - P(pass in 2nd time))*(1 - P(pass in 3rd time))

= 0.22*0.15*0.4 = 0.0132 = 1.3%

1.3% percent of students are not allowed to graduate because of their performance on the exam

(b) percent of students take the test three times = percent of students that fail 2 times

= (percent of students that fail 1st time)*(percent of students that fail 2nd time)

= 22%*15% = 3.3%

3.3% percent of students take the test three times

(c)

percent of students pass a graduation test and receive their high school diplomas :

= (% of students pass in 1st time)+(% of students pass in 2nd time)+(% of students pass in 3rd time)

= (P(pass in 1st time)) + (P(fail in 1st time)*P(pass in 2nd time)) + P(fail in 1st time)*P(fail in 2nd time)*P(pass in 3rd time)

= 0.78 + 0.22*0.85 + 0.22*0.15*0.60

= 0.987 = 98.7%

98.7% percent of students pass a graduation test and receive their high school diplomas

P.S. (please upvote if you find the answer satisfactory)