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Problem-Solving and Data Analysis / Probability and conditional probability Difficulty: Hard
Number of Contestants by Score and Day
 5 out
  of 5		4 out
  of 5		3 out
  of 5		2 out
  of 5		1 out
  of 5		0 out
  of 5		Total
Day 123462320
Day 223554120
Day 333453220
Total7913169660

The same 20 contestants, on each of 3 days, answered 5 questions in order to win a prize. Each contestant received 1 point for each correct answer. The number of contestants receiving a given score on each day is shown in the table above.

No contestant received the same score on two different days. If a contestant is selected at random, what is the probability that the selected contestant received a score of 5 on Day 2 or Day 3, given that the contestant received a score of 5 on one of the three days?

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Explanation

The correct answer is five sevenths. It is given that no contestant received the same score on two different days, so each of the contestants who received a score of 5 is represented in the “5 out of 5” column of the table exactly once. Therefore, the probability of selecting a contestant who received a score of 5 on Day 2 or Day 3, given that the contestant received a score of 5 on one of the three days, is found by dividing the total number of contestants who received a score of 5 on Day 2 or Day 3 2 plus 3, equals 5 by the total number of contestants who received a score of 5, which is given in the table as 7. So the probability is five sevenths. Note that 5/7, .7142, .7143, and 0.714 are examples of ways to enter a correct answer.