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Stanford University EPGY
Math Olympiad
Math Olympiad Problem Solving
Stanford University EPGY Summer Institutes 2008
Problem Set: Probability
1. (2003 AMC 10A #8 and 12A #8) What is the probability that a randomly
drawn positive factor of 60 is less than 7?
(A)
1
10
(B)
1
6
(C)
1
4
(D)
1
3
(E)
1
2
2. (2004 AMC 10B #11) Two eight-sided dice each have faces numbered 1 through
8. When the dice are rolled, each face has an equal probability of appearing on
the top. What is the probability that the product of the two top numbers is
greater than their sum?
(A)
1
2
(B)
47
64
(C)
3
4
(D)
55
64
(E)
7
8
3. (2003 AMC 10A #12) A point (x, y) is picked randomly from inside the rectangle
with vertices (0, 0), (4, 0), (4, 1), and (0, 1). What is the probability that x < y?
(A)
1
8
(B)
1
4
(C)
3
8
(D)
1
2
(E)
3
4
4. (2004 AMC 10B #23 and 12B #20) Each face of a cube is painted either
red or blue, each with probability 1/2. The color of each face is determined
independently. What is the probability that the painted cube can be placed on
a horizontal surface so that the four vertical faces are all the same color?
(A)
1
4
(B)
5
16
(C)
3
8
(D)
7
16
(E)
1
2
5. (2001 AMC 12 #11) A box contains exactly five chips, three red and two white.
Chips are randomly removed one at a time without replacement until all the
red chips are drawn or all the white chips are drawn. What is the probability
that the last chip drawn is white?
(A)
3
10
(B)
2
5
(C)
1
2
(D)
3
5
(E)
7
10
6. (2002 AMC 12B #16) Juan rolls a fair regular eight-sided die. Then Amal rolls
a fair regular six-sided die. What is the probability that the product of the two
rolls is a multiple of 3?
Summer 2008
1
Problem Set: Probability
Stanford University EPGY
(A)
1
12
(B)
Math Olympiad
1
3
(C)
1
2
(D)
7
12
(E)
2
3
7. (2001 AMC 12 #17) A point P is selected at random from the interior of the
pentagon with vertices A = (0, 2), B = (4, 0), C = (2π + 1, 0), D = (2π +
1, 4), E = (0, 4). What is the probability that ∠AP B is obtuse?
(A)
1
5
(B)
1
4
(C)
5
16
(D)
3
8
(E)
1
2
8. (2003 AMC 12B #19) Let S be the set of permutations of the numbers 1,2,3,4,5
for which the first term of the permutation is not 1. A permutation is chosen
randomly from S. The probability that the second term of the chosen permutation is 2, in lowest terms, is a/b. What is a + b?
(A) 5
Summer 2008
(B) 6
(C) 11
2
(D) 16
(E) 19
Problem Set: Probability