AP Statistics · Exam guide
AP Statistics Probability and Random Variables: Expected Value, Binomial, and Normal
Probability questions on the AP Statistics exam come in three common forms: a discrete random variable with a table, a binomial setting, and a normal distribution. Each has a short routine, and graders give credit for naming the model and its parameters. This guide works one example of each, with arithmetic checked.
A discrete random variable
The number of items \(X\) a customer buys has the distribution: \(P(0) = 0.1\), \(P(1) = 0.3\), \(P(2) = 0.4\), \(P(3) = 0.2\). Find the expected value and the standard deviation.
\(E(X) = 0(0.1) + 1(0.3) + 2(0.4) + 3(0.2) = 1.7\). The variance is \[(0 - 1.7)^2(0.1) + (1 - 1.7)^2(0.3) + (2 - 1.7)^2(0.4) + (3 - 1.7)^2(0.2) = 0.81\] so the standard deviation is \(\sqrt{0.81} = 0.9\). On average a customer buys 1.7 items, and the number typically varies from that mean by about 0.9 items.
A binomial setting
A binomial setting has a fixed number of trials \(n\), two outcomes on each trial, independent trials, and the same probability of success \(p\) on each. Always state these conditions and the values of \(n\) and \(p\) before computing.
A basketball player makes 30 percent of free throws, with independent shots. In 10 shots, find the probability of exactly 3 makes, and of at most 3 makes.
Let \(X\) be the number of makes, so \(X\) is binomial with \(n = 10\) and \(p = 0.3\). \(P(X = 3) = \binom{10}{3}(0.3)^3(0.7)^7 \approx 0.267\). \(P(X \le 3) \approx 0.650\). The mean number of makes is \(np = 3\), and the standard deviation is \(\sqrt{np(1-p)} = \sqrt{2.1} \approx 1.449\).
It shows calculator syntax only, with no model, no conditions, and no interpretation.
It names the model and its parameters and shows the setup.
A normal distribution
The heights of adult men in a population are approximately normal with mean 70 inches and standard deviation 4 inches. Find the proportion less than 75 inches.
\(z = \dfrac{75 - 70}{4} = 1.25\). The proportion with \(z \lt 1.25\) is about \(0.894\). So about 89.4 percent are shorter than 75 inches, and about 10.6 percent are taller.
What graders look for
- The model named: "X is binomial with n = ... and p = ..." or "X is normal with mean ... and standard deviation ...".
- The conditions stated and checked in the context.
- A setup that shows the formula or the calculator command with its inputs.
- A sentence that answers the question in the words of the problem.
Common mistakes
- Using the normal model without saying the distribution is approximately normal.
- Using a binomial model when the trials are not independent or the probability changes.
- Giving the standard deviation as the variance, or the reverse.
- Reporting a probability without a sentence in context.
Frequently asked questions
Do I need to state the binomial conditions?
Yes, name them and tie them to the situation, since graders look for it.
What is the difference between variance and standard deviation?
The standard deviation is the square root of the variance and has the same units as the variable.
Which table or calculator function should I use?
Either is acceptable, but show the inputs, such as the mean, standard deviation, and boundary values.
When can a binomial be approximated by a normal?
When the expected numbers of successes and failures are both large, often at least 10. Check the course description.
Where can I find the official scoring rules?
Exam formats change a little from year to year, so check the current course and exam description on the College Board site before test day.
Keep practicing
Every lesson in the AP Statistics course ends with practice problems and an Exam focus box that shows how the topic is tested, and every unit has a 10-problem practice set with worked answers. These free lessons also build the skills the free-response section tests:
See the AP Statistics course Free lessons and lesson list
More AP Statistics guides (8)
Start here
Free-response and writing
- AP Statistics Inference for Proportions Explained: Intervals and Tests with a Worked Example
- AP Statistics Inference for Means Explained: The One-Sample t Test and Checking Conditions
- AP Statistics Chi-Square Test Explained: Homogeneity and Independence with a Worked Example
- AP Statistics Regression Questions Explained: Interpreting Slope, r-Squared and Residuals
- AP Statistics Describing Data: Shape, Center, Spread, and Outliers in Context
- AP Statistics Sampling and Experiments: Bias, Random Samples, and Good Design
Multiple choice
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