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Basic Probability

Sample-space probability: dice, coins, cards, random selection

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Basic ProbabilityAMC 10 Step by Step

Name: Date: October 28, 2026

20 problems, easiest first

What you need to know

Equally likely outcomes. Write $P(E)$ for the probability of an event $E$ . Then $P(E) = \dfrac{\text{outcomes in } E}{\text{all outcomes}}$ , but only when every outcome is truly equally likely. For two dice, count the $36$ ordered pairs, not the $11$ possible sums. Example: a sum of $7$ comes from $6$ of the $36$ pairs, so its probability is $\frac16$ , not $\frac1{11}$ .
The opposite event (complementary counting). $P(E) = 1-P(\text{not } E)$ . This is the key to "at least one" questions. Example: at least one $6$ in $4$ rolls: $1-\left(\frac56\right)^4 = 1-\frac{625}{1296} = \frac{671}{1296}$ .
Independent events multiply. For events that do not affect each other (separate coins or dice, or draws with replacement), $P(A\text{ and }B) = P(A)\cdot P(B)$ . Without replacement, multiply step by step, updating the bag after each draw. Example: $2$ reds in a row from $5$ red and $3$ blue marbles, without replacement: $\frac58\cdot\frac47 = \frac{5}{14}$ .
Common sample spaces (the list of all outcomes): two dice give $36$ , $n$ coins give $2^n$ , and choosing $k$ of $n$ objects gives $\binom{n}{k}$ .
Symmetry. In draws without replacement, each later draw (second, third, …) has the same chances as the first, as long as you don't know what the earlier draws were. Example: the third card dealt from a shuffled deck is an ace with probability $\frac{4}{52}$ , the same as the first card.

How AMC 10 tests it

Problems 4–12: dice, coins, spinners, marbles from a bag, with outcomes few enough to count directly.
Problems 10–18: the probability that a random arrangement or subset has a property, so the real work is the counting underneath.
Problems 12–20: "at least one" or "two the same" questions designed for complementary counting.
Problems 18–25: random whole numbers from $1$ to $n$ with a number theory condition (the product is even, or the numbers are relatively prime, meaning they share no factor bigger than $1$ ), which need casework on remainders.
The choices are fractions in lowest terms, built to catch counts that are off by one.

Standard approaches

Write down the sample space and check that its outcomes are equally likely. Then count the outcomes you want.
If the event says "at least" or "not all the same," find the probability of the opposite and subtract from $1$ .
Multiply along a sequence of draws, shrinking the counts as objects leave the bag. Or use $\binom{n}{k}$ if order does not matter.
Use symmetry: "the first ball is red" and "the last ball is red" have the same probability.
Do casework on the first object or the largest value when the condition depends on it.

Worked example

A bag contains $4$ red, $3$ blue, and $2$ green marbles. Three marbles are drawn at random without replacement. What is the probability that the three marbles are not all of different colors?

(A) $\frac{2}{7}$ (B) $\frac{3}{7}$ (C) $\frac{4}{7}$ (D) $\frac{5}{7}$ (E) $\frac{6}{7}$

Solution. "Not all of different colors" is the opposite of "one of each color," so find the probability of the opposite and subtract it from $1$ .

Step 1: all draws. Order does not matter, so there are $\binom93 = \frac{9\cdot8\cdot7}{3\cdot2\cdot1} = 84$ equally likely draws.

Step 2: one of each color. There are three colors, so we need one red, one blue, and one green: $4\cdot3\cdot2 = 24$ draws.

Step 3: subtract from $1$ . All different colors has probability $\frac{24}{84} = \frac27$ . So "not all different" has probability $1-\frac27 = \frac57$ .

The answer is $\boxed{\textbf{(D)}\ \frac{5}{7}}$ .

Pitfalls

Treating outcomes that are not equally likely as if they were, such as the dice sums $2$ through $12$ .
Forgetting to shrink the bag after each draw without replacement: two reds from $5$ red and $3$ blue is $\frac58\cdot\frac47$ , not $\frac58\cdot\frac58$ .
Counting the outcomes you want in order but the total without order, or the other way around. Pick one way and keep it. For two reds, $\frac{5\cdot4}{\binom82} = \frac{20}{28}$ is wrong; $\frac{5\cdot4}{8\cdot7} = \frac{\binom52}{\binom82} = \frac{5}{14}$ is right.
Answering the opposite of what was asked. Reread the last sentence before you choose.

Traps that recur

Forgetting that rerolling all three dice (15/216) beats keeping a 4 or 5, or failing to count ordered outcomes when listing the favorable rolls. (2020 AMC 10A #25)
Forgetting to multiply the 3 completions by the 3 · 2 choices for the first two moves (giving 3/2187 = 1/729, choice (B)), or using 3^8 as the total. (2006 AMC 10A #25)
Adding up only the straight-up path 0 to 1 to 2 to 3 to 4, giving (1/2)(1/4)^3 = 1/128 and a guess at the smallest choice (A); the backtracking paths are exactly what turn 128 into 97. (2025 AMC 10B #24)
Only forbidding immediate backtracking (probability 125/216) without requiring the path to stay in a unit cube, i.e. that a repeated axis be traversed back the other way. (2024 AMC 10A #24)

1.2023 AMC 10A #7
Janet rolls a standard $6$ -sided die $4$ times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal $3?$
1. A) $\frac{2}{9}$
2. B) $\frac{49}{216}$
3. C) $\frac{25}{108}$
4. D) $\frac{17}{72}$
5. E) $\frac{13}{54}$
2.2022 AMC 10B #12
A pair of fair $6$ -sided dice is rolled $n$ times. What is the least value of $n$ such that the probability that the sum of the numbers face up on a roll equals $7$ at least once is greater than $\frac{1}{2}$ ?
1. A) $2$
2. B) $3$
3. C) $4$
4. D) $5$
5. E) $6$
3.2021 AMC Fall 10A #9
When a certain unfair die is rolled, an even number is $3$ times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even?
1. A) $\dfrac38$
2. B) $\dfrac49$
3. C) $\dfrac59$
4. D) $\dfrac9{16}$
5. E) $\dfrac58$
4.2018 AMC 10B #9
The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6$ . Let $p$ be the probability that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10$ . What other sum occurs with the same probability $p$ ?
1. A) $13$
2. B) $26$
3. C) $32$
4. D) $39$
5. E) $42$
5.2018 AMC 10B #6
A box contains $5$ chips, numbered $1, 2, 3, 4,$ and $5$ . Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$ . What is the probability that $3$ draws are required?
1. A) $\frac{1}{15}$
2. B) $\frac{1}{10}$
3. C) $\frac{1}{6}$
4. D) $\frac{1}{5}$
5. E) $\frac{1}{4}$
6.2017 AMC 10B #9
A radio program has a quiz consisting of $3$ multiple-choice questions, each with $3$ choices. A contestant wins if he or she gets $2$ or more of the questions right. The contestant answers randomly to each question. What is the probability of winning?
1. A) $\frac{1}{27}$
2. B) $\frac{1}{9}$
3. C) $\frac{2}{9}$
4. D) $\frac{7}{27}$
5. E) $\frac{1}{2}$
7.2016 AMC 10A #12
Three distinct integers are selected at random between $1$ and $2016$ , inclusive. Which of the following is a correct statement about the probability $p$ that the product of the three integers is odd?
1. A) $p<\dfrac{1}{8}$
2. B) $p=\dfrac{1}{8}$
3. C) $\dfrac{1}{8}<p<\dfrac{1}{3}$
4. D) $p=\dfrac{1}{3}$
5. E) $p>\dfrac{1}{3}$
8.2016 AMC 10B #12
Two different numbers are selected at random from $( 1, 2, 3, 4, 5)$ and multiplied together. What is the probability that the product is even?
1. A) $0.2$
2. B) $0.4$
3. C) $0.5$
4. D) $0.7$
5. E) $0.8$
9.2013 AMC 10B #12
Let $S$ be the set of sides and diagonals of a regular pentagon. A pair of elements of $S$ are selected at random without replacement. What is the probability that the two chosen segments have the same length?
1. A) $\frac{2}5$
2. B) $\frac{4}9$
3. C) $\frac{1}2$
4. D) $\frac{5}9$
5. E) $\frac{4}5$
10.2012 AMC 10A #9
A pair of six-sided dice are labeled so that one die has only even numbers (two each of $2$ , $4$ , and $6$ ), and the other die has only odd numbers (two each of $1$ , $3$ , and $5$ ). The pair of dice is rolled. What is the probability that the sum of the numbers on the tops of the two dice is $7$ ?
1. A) $\frac{1}{6}$
2. B) $\frac{1}{5}$
3. C) $\frac{1}{4}$
4. D) $\frac{1}{3}$
5. E) $\frac{1}{2}$
11.2011 AMC 10A #14
A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?
1. A) $\frac{1}{36}$
2. B) $\frac{1}{12}$
3. C) $\frac{1}{6}$
4. D) $\frac{1}{4}$
5. E) $\frac{5}{18}$
12.2005 AMC 10A #9
Three tiles are marked $X$ and two other tiles are marked $O$ . The five tiles are randomly arranged in a row. What is the probability that the arrangement reads $XOXOX$ ?
1. A) $\frac{1}{12}$
2. B) $\frac{1}{10}$
3. C) $\frac{1}{6}$
4. D) $\frac{1}{4}$
5. E) $\frac{1}{3}$
13.2005 AMC 10B #15
An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $\$20$ or more?
1. A) ${{{\frac{1}{4}}}}$
2. B) ${{{\frac{2}{5}}}}$
3. C) ${{{\frac{3}{7}}}}$
4. D) ${{{\frac{1}{2}}}}$
5. E) ${{{\frac{2}{3}}}}$
14.2005 AMC 10B #9
One fair die has faces $1, 1, 2, 2, 3, 3$ and another has faces $4, 4, 5, 5, 6, 6.$ The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd?
1. A) $\frac{1}{3}$
2. B) $\frac{4}{9}$
3. C) $\frac{1}{2}$
4. D) $\frac{5}{9}$
5. E) $\frac{2}{3}$
15.2005 AMC 10B #12
Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?
1. A) $\left(\frac{1}{12}\right)^{12}$
2. B) $\left(\frac{1}{6}\right)^{12}$
3. C) $2\left(\frac{1}{6}\right)^{11}$
4. D) $\frac{5}{2}\left(\frac{1}{6}\right)^{11}$
5. E) $\left(\frac{1}{6}\right)^{10}$
16.2004 AMC 10A #10
Coin $A$ is flipped three times and coin $B$ is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same?
1. A) $\frac{19}{128}$
2. B) $\frac{23}{128}$
3. C) $\frac14$
4. D) $\frac{35}{128}$
5. E) $\frac12$
17.2004 AMC 10A #5
A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?
1. A) $\frac{1}{21}$
2. B) $\frac{1}{14}$
3. C) $\frac{2}{21}$
4. D) $\frac{1}{7}$
5. E) $\frac{2}{7}$
18.2025 AMC 10A #14
Six chairs are arranged around a round table. Two students and two teachers randomly select four of the chairs to sit in. What is the probability that the two students will sit in two adjacent chairs and the two teachers will also sit in two adjacent chairs?
1. A) $\frac{1}{6}$
2. B) $\frac{1}{5}$
3. C) $\frac{2}{9}$
4. D) $\frac{3}{13}$
5. E) $\frac{1}{4}$
19.2025 AMC 10B #14
Nine athletes, no two of whom are the same height, try out for the basketball team. One at a time, they draw a wristband at random, without replacement, from a bag containing $3$ blue bands, $3$ red bands, and $3$ green bands. They are divided into a blue group, a red group, and a green group. The tallest member of each group is named the group captain. What is the probability that the group captains are the three tallest athletes?
1. A) $\frac29$
2. B) $\frac 27$
3. C) $\frac9{28}$
4. D) $\frac13$
5. E) $\frac38$
20.2025 AMC 10B #11
On Monday, $6$ students went to the tutoring center at the same time, and each one was randomly assigned to one of the $6$ tutors on duty. On Tuesday, the same $6$ students showed up, the same $6$ tutors were on duty, and the students were again randomly assigned to the tutors. What is the probability that exactly $2$ students met with the same tutor both Monday and Tuesday?
1. A) $\frac{1}{16}$
2. B) $\frac{3}{16}$
3. C) $\frac{1}{4}$
4. D) $\frac{3}{8}$
5. E) $\frac{1}{2}$

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.