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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

16 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)

1.2021 AMC Fall 10B #23
Each of the $5{ }$ sides and the $5{ }$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color?
1. A) $\frac23$
2. B) $\frac{105}{128}$
3. C) $\frac{125}{128}$
4. D) $\frac{253}{256}$
5. E) $1$
2.2019 AMC 10A #20
The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?
1. A) $\dfrac{1}{21}$
2. B) $\dfrac{1}{14}$
3. C) $\dfrac{5}{63}$
4. D) $\dfrac{2}{21}$
5. E) $\dfrac{1}{7}$
3.2016 AMC 10A #17
Let $N$ be a positive multiple of $5$ . One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$ ?
1. A) $12$
2. B) $14$
3. C) $16$
4. D) $18$
5. E) $20$
4.2012 AMC 10A #20
A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90\,^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black?
1. A) $\frac{49}{512}$
2. B) $\frac{7}{64}$
3. C) $\frac{121}{1024}$
4. D) $\frac{81}{512}$
5. E) $\frac{9}{32}$
5.2010 AMC 10A #18
Bernardo randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?
1. A) $\frac{47}{72}$
2. B) $\frac{37}{56}$
3. C) $\frac{2}{3}$
4. D) $\frac{49}{72}$
5. E) $\frac{39}{56}$
6.2009 AMC 10A #24
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?
1. A) $\frac{1}{4}$
2. B) $\frac{3}{8}$
3. C) $\frac{4}{7}$
4. D) $\frac{5}{7}$
5. E) $\frac{3}{4}$
7.2009 AMC 10A #22
Two cubical dice each have removable numbers $1$ through $6$ . 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 $7$ ?
1. A) $\frac{1}{9}$
2. B) $\frac{1}{8}$
3. C) $\frac{1}{6}$
4. D) $\frac{2}{11}$
5. E) $\frac{1}{5}$
8.2009 AMC 10B #25
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the 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?
1. A) $\frac 18$
2. B) $\frac {3}{16}$
3. C) $\frac 14$
4. D) $\frac 38$
5. E) $\frac 12$
9.2008 AMC 10A #22
Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be 6. To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts 1. If it comes up tails, he takes half of the previous term and subtracts 1. What is the probability that the fourth term in Jacob's sequence is an integer?
1. A) $\frac{1}{6}$
2. B) $\frac{1}{3}$
3. C) $\frac{1}{2}$
4. D) $\frac{5}{8}$
5. E) $\frac{3}{4}$
10.2008 AMC 10B #22
Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color?
1. A) $1/12$
2. B) $1/10$
3. C) $1/6$
4. D) $1/3$
5. E) $1/2$
11.2005 AMC 10B #21
Forty slips are placed into a hat, each bearing a number $1$ , $2$ , $3$ , $4$ , $5$ , $6$ , $7$ , $8$ , $9$ , or $10$ , with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all four slips bear the same number. Let $q$ be the probability that two of the slips bear a number $a$ and the other two bear a number $b \neq a$ . What is the value of $q/p$ ?
1. A) $162$
2. B) $180$
3. C) $324$
4. D) $360$
5. E) $720$
12.2004 AMC 10B #23
Each face of a cube is painted either red or blue, each with probability 1/2. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?
1. A) $\frac{1}{4}$
2. B) $\frac{5}{16}$
3. C) $\frac{3}{8}$
4. D) $\frac{7}{16}$
5. E) $\frac{1}{2}$
13.2002 AMC 10A #24
Tina randomly selects two distinct numbers from the set $\{ 1, 2, 3, 4, 5 \}$ , and Sergio randomly selects a number from the set $\{ 1, 2, ..., 10 \}$ . What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?
1. A) $2/5$
2. B) $9/20$
3. C) $1/2$
4. D) $11/20$
5. E) $24/25$
14.2001 AMC 10 #23
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?
1. A) $\frac {3}{10}$
2. B) $\frac {2}{5}$
3. C) $\frac {1}{2}$
4. D) $\frac {3}{5}$
5. E) $\frac {7}{10}$
15.2020 AMC 10A #25
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7.$ Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
1. A) $\frac{7}{36}$
2. B) $\frac{5}{24}$
3. C) $\frac{2}{9}$
4. D) $\frac{17}{72}$
5. E) $\frac{1}{4}$
16.2006 AMC 10A #25
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?
1. A) $\frac{1}{2187}$
2. B) $\frac{1}{729}$
3. C) $\frac{2}{243}$
4. D) $\frac{1}{81}$
5. E) $\frac{5}{243}$

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.