Problem Set 3.1: Probability
Question 1
Question 2
Question 3
Question 4
Question 5
Question 6
Question 1 AMC8 2001 Problem 18
(a) Assuming we are talking about 6-sided dice here… How do you want to start
this question? Get you hands dirty and list all the possible products you can
get. Using a table will make this easier.
(b) Did you list all the possible products? What is the answer?
(c) Can you now see a quicker way to answer this question, without listing all of
the possible outcomes?
(d) If you did this kind of question again, how would you answer it?
(e) Answer the question again, but for multiples of 3.
(f) Answer the question again, but for three dice (multiples of 5).
Question 2 AMC8 2003 Problem 12
(a) Get your hands dirty by getting a dice and drawing out one of the
possibilities. Is the product of the visible faces divisible by 6?
(b) Do we need to draw all of the possible ways the dice can land? Or is there a
more efficient way to think about this question?
(c) If the dice lands with a 6 visible, what can you say?
(d) What if the dice lands with the 6 not visible?
(e) Try this question again, but with the product divisible by 5.
Question 3 AMC8 2000 Problem 21
(a) Get your hands dirty and list all the possible outcomes. How many are there?
How many of these give each person the same number of heads?
(b) Is there a more efficient way to answer this question? How many heads can
they both get?
(c) What is the probability they both get 0 heads? 1 head? 2 head?
(d) What is the probability they get a different number of heads?
(e) Consider a third person with three coins – what is the answer to the question
now?
(f) Generalize the original question so that Keiko has n coins and Ephraim has m
coins. Check your generalization is correct with some specific examples.
Question 4 AMC8 2004 Problem 21
(a) Let’s think… What do you need for two numbers to have an even product?
(b) Do you need to list all the possible outcomes, or can you see a more efficient
way to answer this question? Try and answer the question in the most
efficient way you can think of.
(c) Answer the question again, but with two spinners identical to A.
(d) Answer the question again but with two spinners identical to B.
(e) Can you generalize your result for a spinner A with numbers 1 to m and B
with numbers 1 to n?
(f) Answer the question again, but for the sum of the two spinners not the
product.
(g) Suppose you spin the spinners again, and work out A/B. What is the
probability you get a whole number answer?
Question 5 AMC8 2006 Problem 17
(a) Get your hands dirty: find a possible outcome that has an odd sum.
(b) Generally speaking, in how many different ways can you add three numbers
to get an odd sum? (For example, you can do odd + odd + odd, and…?)
(c) Are all of these ways of making an odd sum possible with these spinners?
Which ways are possible? How does this help you answer the question more
efficiently?
(d) Now answer the question as efficiently as you can.
(e) Answer the question again, but with R changed to all even numbers.
(f) Answer the question again, but with an extra spinner that is a copy of spinner
P.
Question 6 AJHSME 1987 Problem 25
(a) What do we need for the sum of the two numbers to be even?
(b) Be careful here; the ball is not replaced so Jill will not be taking from numbers
1 to 10.
(c) Do you need to list all the possibilities or is there a more efficient way to
answer this question?
(d) Would the answer change if Jack put his ball back in before Jill takes hers? If
so, show how.
(e) Answer the question again, but for 100 balls instead of 10.
(f) Answer the question again, but with a third person taking a third ball.