1997 AMC 8 Problem 20
Attempt Problem 20 of the 1997 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1997 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
20.
A pair of -sided dice has sides numbered through Each side has the same probability (chance) of landing face up. The probability that the product of the two numbers that land face-up exceeds is
Answer: A
Small hint:
For first rolls through count which second rolls make the product exceed
Big hint:
Count ordered pairs from two -sided dice
Solution:
We can case on the value of the first die. If its value is or then it is impossible for the product to be greater than
If it is a then the other die has to roll an otherwise the product is less than
If the first die is a then the other die must be or giving possibilities. If the first die is a then the other die must be or giving possibilities.
Finally, if the first roll is an the other die can be or giving more possibilities.
The total number of working pairs is and the total number of pairs is The desired probability is then
Thus, A is the correct answer.
Problem 20 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
- 1993 AMC 8
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