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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)

1.2024 AMC 10A #17
Two teams are in a best-two-out-of-three playoff: the teams will play at most $3$ games, and the winner of the playoff is the first team to win $2$ games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a $\frac{2}{3}$ chance of winning at home, and its probability of winning when playing away from home is $p$ . Outcomes of the games are independent. The probability that Team A wins the playoff is $\frac{1}{2}$ . Then $p$ can be written in the form $\frac{1}{2}(m - \sqrt{n})$ , where $m$ and $n$ are positive integers. What is $m+n$ ?
1. A) $10$
2. B) $11$
3. C) $12$
4. D) $13$
5. E) $14$
2.2021 AMC 10B #18
A fair $6$ -sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?
1. A) $\frac{1}{120}$
2. B) $\frac{1}{32}$
3. C) $\frac{1}{20}$
4. D) $\frac{3}{20}$
5. E) $\frac{1}{6}$
3.2021 AMC Fall 10A #13
Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls?
1. A) $\frac{1}{64}$
2. B) $\frac{1}{6}$
3. C) $\frac{1}{4}$
4. D) $\frac{5}{16}$
5. E) $\frac{1}{2}$
4.2021 AMC Fall 10B #14
Una rolls $6$ standard $6$ -sided dice simultaneously and calculates the product of the $6{ }$ numbers obtained. What is the probability that the product is divisible by $4?$
1. A) $\frac34$
2. B) $\frac{57}{64}$
3. C) $\frac{59}{64}$
4. D) $\frac{187}{192}$
5. E) $\frac{63}{64}$
5.2020 AMC 10B #11
Ms. Carr asks her students to read any $5$ of the $10$ books on a reading list. Harold randomly selects $5$ books from this list, and Betty does the same. What is the probability that there are exactly $2$ books that they both select?
1. A) $\frac{1}{8}$
2. B) $\frac{5}{36}$
3. C) $\frac{14}{45}$
4. D) $\frac{25}{63}$
5. E) $\frac{1}{2}$
6.2019 AMC 10B #17
A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k=1,2,3,\ldots.$ What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?
1. A) $\frac{1}{4}$
2. B) $\frac{2}{7}$
3. C) $\frac{1}{3}$
4. D) $\frac{3}{8}$
5. E) $\frac{3}{7}$
7.2015 AMC 10B #16
Al, Bill, and Cal will each randomly be assigned a whole number from $1$ to $10$ , inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's?
1. A) $\dfrac{9}{1000}$
2. B) $\dfrac{1}{90}$
3. C) $\dfrac{1}{80}$
4. D) $\dfrac{1}{72}$
5. E) $\dfrac{2}{121}$
8.2014 AMC 10B #16
Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?
1. A) $\frac{1}{36}$
2. B) $\frac{7}{72}$
3. C) $\frac{1}{9}$
4. D) $\frac{5}{36}$
5. E) $\frac{1}{6}$
9.2014 AMC 10A #17
Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?
1. A) $\dfrac16$
2. B) $\dfrac{13}{72}$
3. C) $\dfrac7{36}$
4. D) $\dfrac5{24}$
5. E) $\dfrac29$
10.2008 AMC 10B #16
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, the sum is $0$ .)
1. A) ${{{\frac{3} {8}}}}$
2. B) ${{{\frac{1} {2}}}}$
3. C) ${{{\frac{43} {72}}}}$
4. D) ${{{\frac{5} {8}}}}$
5. E) ${{{\frac{2} {3}}}}$
11.2008 AMC 10B #20
The faces of a cubical die are marked with the numbers $1$ , $2$ , $2$ , $3$ , $3$ , and $4$ . The faces of another die are marked with the numbers $1$ , $3$ , $4$ , $5$ , $6$ , and $8$ . Both dice are thrown. What is the probability that the sum of the top two numbers will be $5$ , $7$ , or $9$ ?
1. A) $5/18$
2. B) $7/18$
3. C) $11/18$
4. D) $3/4$
5. E) $8/9$
12.2008 AMC 10B #17
A poll shows that $70\%$ 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?
1. A) ${{{0.063}}}$
2. B) ${{{0.189}}}$
3. C) ${{{0.233}}}$
4. D) ${{{0.333}}}$
5. E) ${{{0.441}}}$
13.2007 AMC 10B #19
The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by $4,$ and the second number is divided by $5.$ 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?
1. A) $\frac{1}{3}$
2. B) $\frac{4}{9}$
3. C) $\frac{1}{2}$
4. D) $\frac{5}{9}$
5. E) $\frac{2}{3}$
14.2007 AMC 10A #16
Integers $a, b, c,$ and $d$ , not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that $ad-bc$ is even?
1. A) $\frac 38$
2. B) $\frac 7{16}$
3. C) $\frac 12$
4. D) $\frac 9{16}$
5. E) $\frac 58$
15.2006 AMC 10B #17
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?
1. A) $\frac{1}{10}$
2. B) $\frac{1}{6}$
3. C) $\frac{1}{5}$
4. D) $\frac{1}{3}$
5. E) $\frac{1}{2}$
16.2006 AMC 10B #21
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 $1:2:3:4:5:6$ . What is the probability of rolling a total of $7$ on the two dice?
1. A) $\frac{4}{63}$
2. B) $\frac{1}{8}$
3. C) $\frac{8}{63}$
4. D) $\frac{1}{6}$
5. E) $\frac{2}{7}$
17.2004 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?
1. A) $\frac{1}{2}$
2. B) $\frac{47}{64}$
3. C) $\frac{3}{4}$
4. D) $\frac{55}{64}$
5. E) $\frac{7}{8}$
18.2025 AMC 10B #24
A frog hops along the number line according to the following rules.
$\qquad\bullet$ It starts at $0$ .
$\qquad\bullet$ If it is at $0$ , then it moves to $1$ with probability $\tfrac{1}{2}$ and it disappears with probability $\tfrac{1}{2}$ .
$\qquad\bullet$ For $n = 1, 2,$ or $3,$ if it is at $n,$ then it moves to $n+1$ with probability $\tfrac{1}{4},$ it moves to $n-1$ with probability $\tfrac{1}{4},$ and it disappears with $\qquad\phantom{\bullet}$ probability $\tfrac{1}{2}$ .
What is the probability that the frog reaches ${4\,?}$
1. A) $\frac{1}{101}$
2. B) $\frac{1}{100}$
3. C) $\frac{1}{99}$
4. D) $\frac{1}{98}$
5. E) $\frac{1}{97}$
19.2024 AMC 10A #24
A bee is moving in three-dimensional space. A fair six-sided die with faces labeled $A^+, A^-, B^+, B^-, C^+,$ and $C^-$ is rolled. Suppose the bee occupies the point $(a,b,c).$ If the die shows $A^+$ , then the bee moves to the point $(a+1,b,c)$ and if the die shows $A^-,$ then the bee moves to the point $(a-1,b,c).$ Analogous moves are made with the other four outcomes. Suppose the bee starts at the point $(0,0,0)$ and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube?
1. A) $\frac{1}{54}$
2. B) $\frac{7}{54}$
3. C) $\frac{1}{6}$
4. D) $\frac{5}{18}$
5. E) $\frac{2}{5}$
20.2023 AMC 10A #25
If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A, B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$ . For example, $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$ , but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then $d(A, B) = 2$ . Let $Q$ , $R$ , and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of $20$ equilateral triangles). What is the probability that $d(Q, R) > d(R, S)$ ?
1. A) $\frac{7}{22}$
2. B) $\frac{1}{3}$
3. C) $\frac{3}{8}$
4. D) $\frac{5}{12}$
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.