Binomial and Normal Worksheet with Answers – AP Statistics (Free PDF)

Problem 1 Is It Binomial? ★★★

Decide whether each situation is binomial. If not, name the condition that fails.

1. A basketball player takes 15 shots and X is the number made; each shot has the same probability 0.62 and shots are independent.
2. Cards are dealt from a standard deck until a heart appears; X is the number of cards dealt.
3. A teacher randomly picks 5 students from a class of 28 without replacement and X counts those who walk to school.

Problem 2 A Binomial Probability by Formula ★★★

A spinner lands on green with probability 0.4 on each spin. In 6 independent spins, find the probability of exactly 2 greens.

Problem 3 A Cumulative Binomial Probability ★★★

A quiz bowl player answers each question correctly with probability 0.25. In 8 independent questions, find the probability of at most 1 correct answer.

Problem 4 Mean and Standard Deviation of a Count ★★★

In a city, 35% of households own a hybrid car. In a random sample of 60 households, find the mean and standard deviation of the number that own a hybrid.

Problem 5 Reading a Binomial Histogram ★★★

A team wins each of 8 independent games with probability 0.6. The graph shows the distribution of the number of wins X.

Find the most likely number of wins, P(X ≥ 7) and the mean.

Problem 6 At Least One Success ★★★

A seed company claims each seed sprouts with probability 0.85. Five seeds are planted independently. Find the probability that at least one seed does not sprout.

Problem 7 Binomial with a Calculator ★★★

A survey finds that 30% of teenagers use a library card. In a random sample of 15 teenagers, find the probability that at least 7 use a library card. Show the calculator command.

Problem 8 Is the Result Unusual? ★★★

A factory says 10% of its bulbs are defective. In a random sample of 20 bulbs, 5 are defective.

Find P(X ≥ 5). Is the result surprising if the factory’s claim is true?

Problem 9 Normal Area to the Left ★★★

The lengths of adult rainbow trout in a hatchery are approximately normal with mean 14.5 inches and standard deviation 2.1 inches. Find the proportion of trout shorter than 12 inches.

Problem 10 Normal Area to the Right ★★★

Using the same trout distribution (mean 14.5 inches, standard deviation 2.1 inches), find the proportion longer than 18 inches.

Problem 11 Normal Area Between Two Values ★★★

Heights of players in a basketball league are approximately normal with mean 71 inches and standard deviation 4.5 inches.

Find the proportion of players between 66 and 77 inches tall.

Problem 12 Finding a Cutoff ★★★

Scores on a driving test are approximately normal with mean 78 and standard deviation 8. The top 10% of scores earn a bonus. What is the minimum score for the bonus?

Problem 13 Quartiles of a Normal Model ★★★

Weekly spending on snacks is approximately normal with mean $24 and standard deviation $6.50. Find the first and third quartiles.

Problem 14 Comparing Two Normal Models ★★★

Maya scored 672 on a reading test (mean 640, standard deviation 24) and 58 on a math test (mean 52, standard deviation 5). Both distributions are approximately normal. On which test is her percentile higher?

Problem 15 Binomial Probability in Context ★★★

A plant packs 12 batteries in a box. Each battery is defective independently with probability 0.04. Find the probability that a box has at most 2 defective batteries, and the expected number of defective batteries per box.

Problem 16 Surprising Counts ★★★

A bakery claims 30% of its customers order a muffin. In a random sample of 50 customers, 25 order a muffin. Find the mean and standard deviation of the count, how many standard deviations 25 is above the mean, and the exact probability P(X ≥ 25).

Problem 17 Normal Approximation Setup ★★★

In a city, 40% of residents support a new transit tax. A random sample of 150 residents is selected. Verify that a normal approximation is appropriate, then approximate the probability that at least 70 residents in the sample support the tax.

Problem 18 Approximating a Lower Tail ★★★

Of 400 randomly selected packages shipped by a store, each is returned with probability 0.12. The graph shows the distribution of the number returned.

Show that the normal approximation applies and approximate P(X ≤ 40).

Problem 19 Normal Model Then Binomial ★★★

A machine fills juice cups with an amount that is approximately normal with mean 12.1 ounces and standard deviation 0.2 ounce. A cup is underfilled if it contains less than 12 ounces. (a) Find the probability that a cup is underfilled. (b) Eight cups are filled independently; find the probability that at least one is underfilled.

Problem 20 Finding μ and σ from Two Percentiles ★★★

Waiting times at a service desk are approximately normal. Exactly 10% of customers wait less than 40 minutes and exactly 5% wait more than 70 minutes. Find the mean and standard deviation.