Maths Training topic: Probability
AMC 10
1. Two cubical dice each have removable numbers through . The twelve numbers on the two dice
are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the
cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are
added. What is the probability that the sum is ?
2. Three distinct vertices of a cube are chosen at random. What is the probability that the plane
determined by these three vertices contains points inside the cube?
3. Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap
every seconds, and Robert runs clockwise and completes a lap every seconds. Both start
from the start line at the same time. At some random time between minutes and minutes
after they begin to run, a photographer standing inside the track takes a picture that shows onefourth of the track, centered on the starting line. What is the probability that both Rachel and
Robert are in the picture?
4. Each face of a cube is given a single narrow stripe painted from the center of one edge to the
center of its opposite edge. The choice of the edge pairing is made at random and independently
for each face. What is the probability that there is a continuous stripe encircling the cube?
5. Jacob uses the following procedure to write down a sequence of numbers. First he chooses the
first term to be . To generate each succeeding term, he flips a fair coin. If it comes up heads, he
doubles the previous term and subtracts . If it comes up tails, he takes half of the previous term
and subtracts . What is the probability that the fourth term in Jacob's sequence is an integer?
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Done by: Liu Xiaoxu
6. Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What
is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, their sum is .)
7. A poll shows that
of all voters approve of the mayor's work. On three separate occasions a
pollster selects a voter at random. What is the probability that on exactly one of these three
occasions the voter approves of the mayor's work?
8. The faces of a cubical die are marked with the numbers , , , , , and . The faces of a second
cubical die are marked with the numbers , , , , , and . Both dice are thrown. What is the
probability that the sum of the two top numbers will be , , or ?
9. Three red beads, two white beads, and one blue bead are placed in a line in random order. What is
the probability that no two neighboring beads are the same color?
10. Integers , , , and , not necessarily distinct, are chosen independently and at random from to
, inclusive. What is the probability that
is even?
11. The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are
recorded. The first number is divided by , and the second number is divided by . The first
remainder designates a column, and the second remainder designates a row on the checkerboard
shown. What is the probability that the pair of numbers designates a shaded square?
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Done by: Liu Xiaoxu
12. A player pays to play a game. A die is rolled. If the number on the die is odd, the game is lost.
If the number on the die is even, the die is rolled again. In this case the player wins if the second
number matches the first and loses otherwise. How much should the player win if the game is fair?
(In a fair game the probability of winning times the amount won is what the player should pay.)
13. Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red,
and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then
randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that
after this process, the contents of the two bags are the same?
14. For a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5 and 6 on each die are
in the ratio
. What is the probability of rolling a total of 7 on the two dice?
15. Six distinct positive integers are randomly chosen between 1 and 2006, inclusive. What is the
probability that some pair of these integers has a difference that is a multiple of 5?
16. Thee tiles are marked and two other tiles are marked . The five tiles are randomly arranged
in a row. What is the probability that the arrangement reads
?
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Done by: Liu Xiaoxu
17. One fair die has faces , , , , , and another has faces , , , , , . The dice are rolled and
the numbers on the top faces are added. What is the probability that the sum will be odd?
18. Twelve fair dice are rolled. What is the probability that the product of the numbers on the top
faces is prime?
19. An envelope contains eight bills: ones, fives, tens, and twenties. Two bills are drawn at
random without replacement. What is the probability that their sum is
or more?
20. Forty slips are placed into a hat, each bearing a number , , , , , , , , , or , with each
number entered on four slips. Four slips are drawn from the hat at random and without
replacement. Let be the probability that all four slips bear the same number. Let be the
probability that two of the slips bear a number and the other two bear a number
. What is
the value of
?
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Done by: Liu Xiaoxu
AMC 12
1. Each face of a cube is given a single narrow stripe painted from the center of one edge to the
center of its opposite edge. The choice of the edge pairing is made at random and independently
for each face. What is the probability that there is a continuous stripe encircling the cube?
2. Two circles of radius 1 are to be constructed as follows. The center of circle
is chosen
uniformly and at random from the line segment joining
and
. The center of circle is
chosen uniformly and at random, and independently of the first choice, from the line segment
joining
to
. What is the probability that circles
and
intersect?
3. Integers
and not necessarily distinct, are chosen independently and at random from to
inclusive. What is the probability that
is even?
4. A bug starts at one vertex of a cube and moves along the edges of the cube according to the
following rule. At each vertex the bug will choose to travel along one of the three edges
emanating from that vertex. Each edge has equal probability of being chosen, and all choices are
independent. What is the probability that after seven moves the bug will have visited every vertex
exactly once?
5. On a standard die one of the dots is removed at random with each dot equally likely to be chosen.
The die is then rolled. What is the probability that the top face has an odd number of dots?
6. Select numbers and between and independently and at random, and let be their sum. Let
and be the results when
and , respectively, are rounded to the nearest integer. What
is the probability that
?
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Done by: Liu Xiaoxu
7. What is the probability that a randomly drawn positive factor of
than ?
8. Tina randomly selects two distinct numbers from the set
is less
and Sergio randomly
selects a number from the set
. The probability that Sergio's number is larger than
the sum of the two numbers chosen by Tina is
9. 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?
10. Professor Gamble buys a lottery ticket, which requires that he pick six different integers from
through , inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his
six numbers is an integer. It so happens that the integers on the winning ticket have the same
property— the sum of the base-ten logarithms is an integer. What is the probability that Professor
Gamble holds the winning ticket?
11. Six points on a circle are given. Four of the chords joining pairs of the six points are selected at
random. What is the probability that the four chords are the sides of a convex quadrilateral?
12. Two six-sided dice are fair in the sense that each face is equally likely to turn up. However, one
of the dice has the replaced by and the other die has the replaced by . When these dice are
rolled, what is the probability that the sum is an odd number?
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Done by: Liu Xiaoxu
13. A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses
equal the third, what is the probability that at least one "2" is tossed?
14. If
and
are three (not necessarily different) numbers chosen randomly and with replacement
from the set
, the probability that
is even is
A.
B.
C.
D.
E.
15. First is chosen at random from the set
, and then is chosen at random
from the same set. The probability that the integer
has units digit
is
16. Let be a real number selected uniformly at random between 100 and 200. If
the probability that
.(
, find
means the greatest integer less than or equal to .)
17. One student in a class of boys and girls is chosen to represent the class. Each student is equally
likely to be chosen and the probability that a boy is chosen is of the probability that a girl is
chosen. The ratio of the number of boys to the total number of boys and girls is
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Done by: Liu Xiaoxu
18. A box contains 11 balls, numbered 1,2,3,....,11. If 6 balls are drawn simultaneously at random,
what is the probability that the sum of the numbers on the balls drawn is odd?
A.
B.
C.
D.
E.
19. Three balls marked 1, 2, and 3, are placed in an urn. One ball is drawn, its number is recorded,
then the ball is returned to the urn. This process is repeated and then repeated once more, and
each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is 6,
what is the probability that the ball numbered 2 was drawn all three times?
20. The adjacent map is part of a city: the small rectangles are rocks, and the paths in between are
streets. Each morning, a student walks from intersection A to intersection B, always walking
along streets shown, and always going east or south. For variety, at each intersection where he
has a choice, he chooses with probability whether to go east or south. Find the probability that
through any given morning, he goes through .
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Done by: Liu Xiaoxu