This precalculus counting probability worksheet gives 25 problems on counting, permutations, combinations and probability, ordered from easy to hard, with a separate answer key.
These 25 problems practice the counting principle, permutations, combinations, probability with counting, the general multiplication rule and expected value (S-CP.B.8, S-CP.B.9, S-MD.A.1, S-MD.A.2). They move from basic counts to multi-step word problems.
Write each count as a product before evaluating, and keep probabilities as exact fractions.
Precalculus Counting Probability Worksheet: problems 1–5
Problem 1: Fundamental counting principle
Find the number of possibilities.
- Ethan can choose from 4 shirts, 3 pairs of pants and 2 pairs of shoes. How many outfits are possible?
- A sandwich shop offers 5 breads, 3 meats and 4 toppings, and a sandwich has one of each. How many sandwiches are possible?
Problem 2: Factorials
Evaluate each expression.
- \(6!\)
- \(\frac{7!}{5!}\)
- \(\frac{8!}{3!\,5!}\)
Problem 3: PIN codes
A phone PIN has 4 digits chosen from 0 through 9, as in the boxes below.
- How many PINs are possible if digits may repeat?
- How many PINs are possible if no digit may repeat?
Problem 4: Permutations
Evaluate each permutation.
- \(_{8}P_{3}\)
- \(_{10}P_{2}\)
Problem 5: Combinations
Evaluate each combination.
- \(_{10}C_{4}\)
- \(_{12}C_{10}\)
Precalculus Counting Probability Worksheet: problems 6–10
Problem 6: Arranging letters
Find the number of distinct arrangements of all the letters in each word.
- MATH
- BANANA
- STATISTICS
Problem 7: Order or no order
Decide whether each situation is a permutation or a combination, then calculate.
- Choose 3 of 12 students to attend a conference.
- Award gold, silver and bronze medals among 12 runners.
Problem 8: Handshakes
Six friends meet and each person shakes hands once with every other person.
How many handshakes happen?
Problem 9: Probability with two dice
Two fair dice are rolled. The grid shows all sums.
- Find the probability that the sum is 8.
- Find the probability that the sum is at least 10.
Problem 10: Probability with cards
Two cards are dealt from a standard 52-card deck.
- Find the probability that both are hearts.
- Find the probability that both are face cards (jack, queen or king).
Precalculus Counting Probability Worksheet: problems 11–15
Problem 11: Choosing a committee
A class has 10 girls and 8 boys. Four students are chosen at random for a committee. Find the probability that all four are girls.
Problem 12: Drawing without replacement
Two marbles are drawn, one after the other and without replacement, from the bag below.
- Find the probability that both are red.
- Find the probability that the first is red and the second is blue.
Problem 13: Using the multiplication rule
Use the general multiplication rule.
- \(P(A)=0.3\) and \(P(B\mid A)=0.6\). Find \(P(A\text{ and }B)\).
- \(P(A\text{ and }B)=0.24\) and \(P(A)=0.5\). Find \(P(B\mid A)\).
Problem 14: At least one
Use the complement to find each probability.
- A fair coin is flipped 3 times . Find the probability of at least one head.
- A fair die is rolled 3 times . Find the probability of at least one six.
Problem 15: Probability distribution
Let \(X\) be the number of heads when a fair coin is flipped 3 times . Complete the table, then find \(E(X)\).
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(X = x) | ? | ? | ? | ? |
Precalculus Counting Probability Worksheet: problems 16–20
Problem 16: Missing probability
A random variable \(X\) has the distribution below.
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(X = x) | 0.1 | 0.3 | p | 0.2 |
- Find \(p\).
- Find \(E(X)\).
- Find \(P(X\ge 2)\).
Problem 17: Free throws
Priya makes 80% of her free throws. She shoots two free throws, and the shots are independent.
- Find the probability that she makes both.
- Find the probability that she makes exactly one.
- Find the expected number of shots made.
Problem 18: Books on a shelf
Noah arranges 5 different books on a shelf.
- How many arrangements are possible?
- How many arrangements keep two particular books next to each other?
Problem 19: Student council
A school has 6 girls and 7 boys running for a 5-person committee.
- How many committees have exactly 2 girls and 3 boys?
- If the committee is chosen at random from all 13 students, find the probability it has exactly 2 girls and 3 boys.
Problem 20: License plates
A state plate has 3 letters followed by 3 digits.
- How many plates are possible if letters and digits may repeat?
- How many are possible if no letter and no digit may repeat?
Precalculus Counting Probability Worksheet: problems 21–25
Problem 21: Raffle
A school raffle sells 200 tickets at $5 each. The prizes are one $300 prize, three $50 prizes and six $10 prizes. Find the expected net result for a person who buys one ticket.
Problem 22: Insurance
Aisha’s company charges a $250 yearly premium for a policy. There is a 2% chance a customer files a $5,000 claim. Find the company’s expected profit per policy.
Problem 23: Five hearts
A 5-card hand is dealt from a standard deck. Find the probability that all five cards are hearts, as a fraction in lowest terms and as a decimal rounded to 6 places.
Problem 24: Two-stage tree
Liam draws two marbles without replacement from a jar with 4 green and 6 yellow marbles. Find the probability that the two marbles are the same color.
Problem 25: A dice game
A carnival game costs $2.50 to play. Sofia rolls one fair die and wins $12 for a six, $3 for a four or five, and nothing otherwise. Find the expected net result and say whether the game favors the player.
How to use this precalculus counting probability worksheet
Print this precalculus counting probability worksheet as a PDF, solve every problem on paper, then compare with the answer key.
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