StatisticsdrillProbability practice problems, with answers checked as you go
Probability is the part of a statistics course where reading the chapter does not help much: you have to work problems until the setups become automatic. These problems are arranged in the order a first course teaches them, from basic rules through conditional probability and the named distributions. Type your answer, see at once whether it is right, then see the formula that gets there.
Basic probability rules
Start here if "at least one" problems still trip you up. The complement rule, the addition rule for overlapping events and the multiplication rule for independent events cover most of what a first exam asks.
Conditional probability and Bayes' theorem
Conditional probability problems are easiest from a table: count the cases that satisfy the condition, then count how many of those also satisfy the event. Bayes' theorem is the same idea with the condition flipped, which is why the medical test problem feels backwards the first time.
Counting: permutations and combinations
Order matters for a permutation (a ranking, a password) and not for a combination (a committee, a hand of cards). The hypergeometric problem combines two combinations in one fraction, which is how lottery and quality-control odds are computed.
Binomial distribution problems
A binomial setup needs a fixed number of trials, two outcomes and a constant probability. "Exactly k" uses the formula once; "at least k" means adding several terms or using the complement. The mean is np and the standard deviation is the square root of np(1-p).
Poisson distribution problems
Poisson is the distribution of counts in a fixed interval: calls per hour, defects per meter. The only parameter is the average rate, and "at most k" means summing the terms from zero to k.
Normal distribution problems
Every normal distribution problem is a z-score problem: standardize, then read the area. The three shapes are "between two values", "above a value" and the reverse question, "which value cuts off the top 10 percent". The sampling distribution of the mean uses the same steps with the standard error in place of the standard deviation.
How the problems are checked
Each problem has a numeric answer with a tolerance, so rounding to the number of decimals the question asks for is enough. Wrong answers get a diagnosis where there is a common one (the complement taken the wrong way, order counted when it should not be), and after two attempts the full working opens. Timed mode puts a clock on each problem for exam practice.
Work the problems
Five problems are free without an account. Pro unlocks all 55, with worked solutions and progress saved to your account.
Start practicing freeQuestions
Are these probability problems suitable for AP Statistics and intro college courses?
Yes. The set follows the order of a first course: probability rules, conditional probability and Bayes, counting, binomial, Poisson, the normal distribution and sampling distributions, then inference.
Do I need a calculator?
A basic calculator is enough. Binomial, Poisson and normal probabilities can be computed with the formulas shown, a scientific calculator, or a spreadsheet function such as NORM.DIST or BINOM.DIST.
How precise does my answer need to be?
Each problem states the decimals it expects and accepts answers within a small tolerance, so standard rounding passes.
Are solutions shown?
After two attempts, or straight away for solved problems, the full working opens with the formula and each step. In timed mode solutions stay closed until the clock runs out.