Under the \"no pass, no play\" rule for athletes, an athlete who fails a course

Question

Under the "no pass, no play" rule for athletes, an athlete who fails a course is disqualified from participating in sports activities during the next grading period. Suppose the probability that an athlete who has not previously been disqualified will be disqualified is 0.14 and the probability that an athlete who has been disqualified will be disqualified again in the next time period is 0.5. If 30% of the athletes have been disqualified before, what is the unconditional probability that an athlete will be disqualified during the next grading period? (Enter your answer to three decimal places.)

Explanation / Answer

Suppose there are 100 students altogether
30% disqualified = 30 students got disqualified
chance for them to get disqualified in the next grading period is 50%.
Therefore there is a chance of 14 students to be disqualified out of the thirty students.

70 students passed through the grading session
Chance for them to be disqualified for the next grading session is 14%.
So there is a chance of 70*0.14 = 9.8 to be disqualified.

The student can either from the 70 students that got passed the grading or among the 30 students who got disqualified.

So total probability = 14 + 9.8 = 23.8 % = 0.238

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