Probability Practice Questions: Test Your Knowledge | LearnByTeaching.ai
These 40 probability practice questions cover counting principles, conditional probability, random variables, and probability distributions. They range from introductory combinatorics to challenging applications of Bayes' theorem and expected value, helping you identify areas that need more review.
40 questions total
Counting Principles and Basic Probability
Covers permutations, combinations, sample spaces, and fundamental probability rules.
How many ways can you arrange 5 distinct books on a shelf?
A fair six-sided die is rolled. What is the probability of rolling an even number?
How many ways can you choose 3 students from a group of 10 to form a committee?
Two fair coins are flipped. What is the probability of getting at least one head?
A password consists of 3 digits (0-9) followed by 2 uppercase letters (A-Z). How many distinct passwords are possible?
In how many ways can 8 people be seated in a row if two specific people must sit next to each other?
If P(A) = 0.4 and P(B) = 0.3, and A and B are mutually exclusive, what is P(A or B)?
A bag contains 5 red and 7 blue marbles. Two marbles are drawn without replacement. What is the probability both are red?
How many distinct arrangements are there of the letters in the word MISSISSIPPI?
A class has 12 boys and 8 girls. A team of 5 is chosen randomly. What is the probability the team has exactly 3 boys and 2 girls?
Conditional Probability and Bayes' Theorem
Covers conditional probability, independence, Bayes' theorem, and the law of total probability.
If P(A) = 0.6, P(B) = 0.5, and P(A and B) = 0.3, what is P(A|B)?
Events A and B are independent with P(A) = 0.4 and P(B) = 0.3. What is P(A and B)?
A test for a disease has 95% sensitivity (true positive rate) and 90% specificity (true negative rate). The disease prevalence is 1%. What is the probability a person with a positive test actually has the disease?
Two cards are drawn from a standard 52-card deck without replacement. What is the probability the second card is a king given the first card is a king?
In a class, 60% study math, 40% study science, and 20% study both. What fraction of science students also study math?
A factory has three machines producing widgets. Machine A produces 50% of widgets with 2% defect rate, B produces 30% with 3% defect rate, and C produces 20% with 5% defect rate. A widget is found defective. What is the probability it came from Machine C?
If P(A|B) = P(A), which of the following must be true?
You roll two dice. Given that the sum is at least 10, what is the probability the sum is exactly 12?
Three fair coins are tossed. What is the probability of getting all heads given that at least two are heads?
Events A and B satisfy P(A) = 0.3, P(B|A) = 0.8, and P(B|A') = 0.2, where A' is the complement of A. What is P(B)?
Random Variables and Expected Value
Covers discrete and continuous random variables, probability mass functions, expected value, and variance.
A random variable X has the distribution: P(X=1) = 0.3, P(X=2) = 0.5, P(X=3) = 0.2. What is E(X)?
If you bet $1 on a fair coin flip and win $2 on heads or lose $1 on tails, what is your expected profit per flip?
If X and Y are independent random variables with Var(X) = 4 and Var(Y) = 9, what is Var(X + Y)?
A fair die is rolled. Let X be the result. What is Var(X)?
If E(X) = 5 and E(Y) = 3, what is E(2X - 3Y + 7)?
What is the expected number of rolls of a fair die needed to get the first 6?
A random variable X follows a Poisson distribution with mean 4. What is P(X = 0)?
If X is uniformly distributed on [0, 10], what is P(3 < X < 7)?
Two independent random variables X and Y each have mean 0 and variance 1. What is E((X + Y)^2)?
A game costs $5 to play. You draw a card from a standard deck: face card wins $15, ace wins $25, anything else wins nothing. What is the expected profit?
Probability Distributions
Covers binomial, normal, geometric, and other key distributions with their properties and applications.
A coin with P(heads) = 0.6 is flipped 5 times. What is the probability of getting exactly 3 heads?
For a standard normal distribution, approximately what percentage of values fall within one standard deviation of the mean?
If X ~ Binomial(n=20, p=0.5), what are the mean and standard deviation of X?
A z-score of 2.0 corresponds to a value that is:
The number of customers arriving at a store follows a Poisson distribution with an average of 10 per hour. What is the probability of exactly 10 customers in one hour?
Which distribution models the number of trials until the first success in independent Bernoulli trials?
X ~ Normal(100, 15). What is P(X > 130)?
A binomial distribution with n=100 and p=0.5 can be well approximated by which normal distribution?
If X has an exponential distribution with rate lambda = 2, what is E(X)?
The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as sample size increases, regardless of the population distribution. This requires:
Scoring Guide
Total possible: 40
Study Recommendations
- Draw tree diagrams for every conditional probability problem to visualize the branching outcomes
- Practice the complement technique — whenever you see 'at least one,' compute 1 minus the probability of none
- For counting problems, always ask: does order matter? Is repetition allowed? This determines permutations vs. combinations
- Simulate probability experiments in code to build intuition before doing the math
- Master the key distribution formulas (binomial, Poisson, normal) and know when each applies
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