🎲 Counting-Probability Warmups
Recommended: 30–40 minutes. No calculator.
Problems
1.
Tags: Basic Counting · Easy · source: Original (AMC-style)
How many ways can you arrange 3 books on a shelf?
A) $3$ B) $6$ C) $9$ D) $12$ E) $27$
Answer & Solution
Answer: B
This is $3! = 3 \times 2 \times 1 = 6$ ways.
2.
Tags: Combinations · Easy · source: Original (AMC-style)
How many ways can you choose 2 people from 4 people?
A) $4$ B) $6$ C) $8$ D) $12$ E) $16$
Answer & Solution
Answer: B
This is $C(4,2) = \frac{4!}{2!2!} = \frac{24}{4} = 6$.
3.
Tags: Basic Probability · Easy · source: Original (AMC-style)
What is the probability of rolling a 3 on a fair die?
A) $\frac{1}{6}$ B) $\frac{1}{3}$ C) $\frac{1}{2}$ D) $\frac{2}{3}$ E) $\frac{5}{6}$
Answer & Solution
Answer: A
There is 1 favorable outcome out of 6 total outcomes, so $P = \frac{1}{6}$.
4.
Tags: Coin Toss · Easy · source: Original (AMC-style)
What is the probability of getting heads on a fair coin?
A) $\frac{1}{4}$ B) $\frac{1}{3}$ C) $\frac{1}{2}$ D) $\frac{2}{3}$ E) $\frac{3}{4}$
Answer & Solution
Answer: C
There is 1 favorable outcome out of 2 total outcomes, so $P = \frac{1}{2}$.
5.
Tags: Permutations · Easy · source: Original (AMC-style)
How many ways can you arrange the letters in ‘ABC’?
A) $3$ B) $6$ C) $9$ D) $12$ E) $27$
Answer & Solution
Answer: B
This is $3! = 6$ ways.
6.
Tags: Complementary Probability · Easy · source: Original (AMC-style)
What is the probability of NOT rolling a 1 on a fair die?
A) $\frac{1}{6}$ B) $\frac{1}{3}$ C) $\frac{1}{2}$ D) $\frac{2}{3}$ E) $\frac{5}{6}$
Answer & Solution
Answer: E
P(not 1) = 1 - P(1) = $1 - \frac{1}{6} = \frac{5}{6}$.
7.
Tags: Independent Events · Easy · source: Original (AMC-style)
What is the probability of getting heads twice in a row?
A) $\frac{1}{4}$ B) $\frac{1}{3}$ C) $\frac{1}{2}$ D) $\frac{2}{3}$ E) $\frac{3}{4}$
Answer & Solution
Answer: A
P(heads and heads) = P(heads) × P(heads) = $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.
8.
Tags: Basic Counting · Easy · source: Original (AMC-style)
How many ways can you choose 1 item from 5 items?
A) $1$ B) $3$ C) $5$ D) $10$ E) $15$
Answer & Solution
Answer: C
This is $C(5,1) = 5$ ways.
9.
Tags: Probability Range · Easy · source: Original (AMC-style)
What is the probability of an impossible event?
A) $0$ B) $\frac{1}{2}$ C) $1$ D) $2$ E) Cannot be determined
Answer & Solution
Answer: A
An impossible event has probability $0$.
10.
Tags: Certain Event · Easy · source: Original (AMC-style)
What is the probability of a certain event?
A) $0$ B) $\frac{1}{2}$ C) $1$ D) $2$ E) Cannot be determined
Answer & Solution
Answer: C
A certain event has probability $1$.
11.
Tags: Basic Combinations · Easy · source: Original (AMC-style)
How many ways can you choose 0 items from 5 items?
A) $0$ B) $1$ C) $5$ D) $10$ E) $25$
Answer & Solution
Answer: B
There is exactly 1 way to choose nothing: $C(5,0) = 1$.
12.
Tags: Factorial · Easy · source: Original (AMC-style)
What is $4!$?
A) $12$ B) $16$ C) $20$ D) $24$ E) $32$
Answer & Solution
Answer: D
$4! = 4 \times 3 \times 2 \times 1 = 24$.
13.
Tags: Basic Probability · Easy · source: Original (AMC-style)
What is the probability of rolling an even number on a fair die?
A) $\frac{1}{6}$ B) $\frac{1}{3}$ C) $\frac{1}{2}$ D) $\frac{2}{3}$ E) $\frac{5}{6}$
Answer & Solution
Answer: C
There are 3 even numbers (2,4,6) out of 6 total, so $P = \frac{3}{6} = \frac{1}{2}$.
14.
Tags: Counting Principle · Easy · source: Original (AMC-style)
If you have 3 shirts and 2 pants, how many outfits can you make?
A) $5$ B) $6$ C) $8$ D) $10$ E) $12$
Answer & Solution
Answer: B
Using the multiplication principle: $3 \times 2 = 6$ outfits.
15.
Tags: Basic Permutations · Easy · source: Original (AMC-style)
How many ways can you arrange 2 letters from ‘ABCD’?
A) $4$ B) $6$ C) $8$ D) $12$ E) $16$
Answer & Solution
Answer: D
This is $P(4,2) = 4 \times 3 = 12$ ways.
Answer Key
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ans | B | B | A | C | B | E | A | C | A | C | B | D | C | B | D |