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## Dicey Die

Consider this problem from the 1995 American Junior High school Mathematics Exam (AJHSME):

Diana and Apollo each roll a standrad die obtaining a number at random from 1 to 6. What is the probability that Diana's number is larger than Apollo's number?

• (A) 1/3
• (B) 5/12
• (C) 4/9
• (D) 17/36
• (E) 1/2
Note: Given the results of the 241,180 exams taken that year, this was the most difficult problem "statistically" in that only 15.31% of the junior high school students answered it correctly.

Hint: Get two dice and a friend...and then perform the experiment multiple times. What does your experimental data tell you? What possible comparisons do you constantly make after each roll?

Solution Commentary: Set up a 6x6 table showing all of the 36 possibilities of the two rolls. Shade in those cells where the roll for one specific player exceeds the other player's rolls.

You should see an interesting visual pattern related to the sum 1+2+3+...., which will help produce a final answer.

Now, the real problem: Why were so few of the middle school students able to solve this problem?