This ap statistics binomial normal worksheet gives 24 problems on binomial and normal distributions, ordered from easy to hard, with a separate answer key.
These problems practice the two main models of AP Statistics Unit 2 (2026-27): the binomial distribution and the normal distribution. You will check the BINS conditions, find binomial probabilities, means and standard deviations, and use z-scores and areas under the normal curve.
The last problems combine skills, including normal approximations to binomial counts and judging whether a result is unusual. Show your setup, and round probabilities to four decimal places.
AP Statistics Binomial Normal Worksheet: problems 1–5
Problem 1: Coin flips
Jamal flips a fair coin \(12\) times and lets \(X\) be the number of heads. Decide whether \(X\) is a binomial random variable. If it is, state \(n\) and \(p\).
Problem 2: Cards without replacement
Sofia deals \(10\) cards from a well-shuffled standard \(52\)-card deck without putting any back. Let \(X\) be the number of hearts. Explain why \(X\) is not binomial.
Problem 3: Rolling until a six
Ethan rolls a fair die until a \(6\) appears and lets \(X\) be the number of rolls. Explain why \(X\) is not binomial.
Problem 4: Exactly three successes
A multiple-choice quiz has \(8\) questions, each with four choices. Liam guesses on every question, so \(X\sim\) binomial with \(n=8\) and \(p=0.25\). Find \(P(X=3)\).
Problem 5: No successes
Priya plants \(6\) seeds that each sprout with probability \(0.9\). Let \(X\) be the number that fail to sprout, so \(p=0.1\) for a failure. Find the probability that no seed fails.
AP Statistics Binomial Normal Worksheet: problems 6–10
Problem 6: At most two
Let \(X\) be binomial with \(n=5\) and \(p=0.4\). Find \(P(X\le 2)\).
Problem 7: At least one
A machine makes bolts and \(5\%\) are defective, independently. Mei checks \(10\) bolts. Find the probability that at least one is defective.
Problem 8: Mean and standard deviation
A binomial random variable has \(n=40\) and \(p=0.35\). Find its mean and standard deviation.
Problem 9: Interpreting the mean
A factory knows that \(2\%\) of phone cases are cracked. A shipment of \(200\) is inspected. Find the mean and standard deviation of the number of cracked cases, and interpret the mean in context.
Problem 10: Reading a distribution graph
The graph shows the distribution of \(X\), the number of heads in \(4\) fair coin flips. Use the graph to find \(P(X\ge 3)\).
AP Statistics Binomial Normal Worksheet: problems 11–15
Problem 11: Probability of a middle range
Let \(X\) be binomial with \(n=8\) and \(p=0.5\), graphed below. Find \(P(3\le X\le 5)\).
Problem 12: z-score of a height
Heights of adult men in a town are approximately normal with mean \(69\) inches and standard deviation \(3\) inches. Find the z-score of a man who is \(75\) inches tall and interpret it.
Problem 13: Area to the right
Scores on a college entrance exam are approximately normal with mean \(500\) and standard deviation \(100\). Find the proportion of students who score above \(620\).
Problem 14: Area between two values
Pizza delivery times in a city are approximately normal with mean \(72\) minutes and standard deviation \(6\) minutes. Find the probability that an order arrives between \(65\) and \(80\) minutes.
Problem 15: Finding a percentile
IQ scores are approximately normal with mean \(100\) and standard deviation \(15\). What score marks the 90th percentile?
AP Statistics Binomial Normal Worksheet: problems 16–20
Problem 16: Empirical rule
Bags of flour are filled with a mean of \(16.0\) ounces and a standard deviation of \(0.2\) ounce, with a normal distribution. Use the 68-95-99.7 rule to find the percent of bags that (a) weigh between \(15.6\) and \(16.4\) ounces, and (b) weigh less than \(15.8\) ounces.
Problem 17: Comparing z-scores
Maya scored \(84\) on a test with mean \(75\) and standard deviation \(6\). Diego scored \(88\) on a different test with mean \(80\) and standard deviation \(8\). Who did better relative to the class?
Problem 18: Checking the approximation
Decide whether the normal approximation to the binomial is reasonable for (a) \(n=150\), \(p=0.2\), and (b) \(n=40\), \(p=0.1\). Show the check.
Problem 19: Normal approximation
Let \(X\) be binomial with \(n=100\) and \(p=0.3\). (a) Check the conditions for a normal approximation. (b) Use the normal model to approximate \(P(X\ge 35)\). (c) The exact binomial value is \(0.1629\). Why is the approximation smaller?
Problem 20: Free throws
Priya makes \(70\%\) of her free throws, independently. She shoots \(15\) free throws. Find the probability that she makes at least \(12\).
AP Statistics Binomial Normal Worksheet: problems 21–24
Problem 21: Overbooked flight
An airline sells \(54\) tickets for a flight with \(50\) seats. Each ticketed passenger shows up with probability \(0.9\), independently. Find the mean and standard deviation of the number who show up, and the probability that more than \(50\) show up.
Problem 22: Recovering n and p
A binomial random variable has mean \(24\) and variance \(9.6\). Find \(n\) and \(p\).
Problem 23: Poll approximation
In a city, \(45\%\) of voters support a bond issue. A pollster surveys \(200\) randomly chosen voters (far fewer than \(10\%\) of the city). Use a normal approximation to find the probability that more than \(100\) of the sampled voters support it.
Problem 24: Is the claim believable?
A manufacturer claims that only \(3\%\) of its light bulbs are defective. An inspector tests \(150\) randomly chosen bulbs and finds \(10\) defective. (a) Find the mean and standard deviation of the number of defective bulbs if the claim is true. (b) Can a normal model be used to find \(P(X\ge 10)\)? (c) Find \(P(X\ge 10)\) and decide whether the result casts doubt on the claim.
How to use this ap statistics binomial normal worksheet
Print this ap statistics binomial normal worksheet as a PDF, solve every problem on paper, then compare with the answer key.
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