AP Statistics: Probability and Distributions

Probability and distributions sit at the core of AP Statistics because they connect everyday uncertainty to precise mathematical models. Whether you are judging the reliability of a manufacturing process, predicting the number of customers who arrive in an hour, or interpreting a poll, you are using probability to describe what could happen and distributions to describe how likely each outcome is.

This article focuses on the essentials: probability rules, conditional probability, random variables (discrete and continuous), common distributions (binomial and normal), combining random variables, and sampling distributions.

Probability: The Rules That Keep You Honest

Probability is a number between 0 and 1 that measures how likely an event is. In AP Statistics, you typically work with events, which are sets of outcomes from some random process.

Basic probability rules

A few rules show up constantly:

• Complement: If $A$ is an event, then $P(A^c)=1-P(A)$.
• Addition rule: For any events $A$ and $B$,

$$P(A\cup B)=P(A)+P(B)-P(A\cap B).$$ If $A$ and $B$ are mutually exclusive (disjoint), then $P(A\cap B)=0$, so you can simply add.

• Multiplication rule:

$$P(A\cap B)=P(A)P(B|A).$$ This is the foundation for multi-step probability calculations.

A common AP-level skill is translating a word scenario into one of these structures. “At least one” often suggests complements. “Either A or B” suggests the addition rule. “Both” suggests intersection and the multiplication rule.

Conditional probability and independence

Conditional probability updates the chance of an event after you learn something else happened: $$P(A|B)=\frac{P(A\cap B)}{P(B)}\text{(when }P(B)>0\text{)}.$$

Independence is stronger than “not related” in a casual sense. Events $A$ and $B$ are independent if knowing $B$ does not change the probability of $A$: $$P(A|B)=P(A).$$ Equivalently, $$P(A\cap B)=P(A)P(B).$$

In practice, many mistakes come from assuming independence when it is not warranted. Sampling without replacement, for example, usually creates dependence because earlier selections change what is left.

Random Variables: Turning Outcomes Into Numbers

A random variable assigns a numerical value to each outcome of a random process. That numerical viewpoint allows you to compute expected values, standard deviations, and probabilities using distributions.

Discrete random variables

A discrete random variable takes on countable values (often integers), such as the number of heads in 10 coin flips. It is described by a probability distribution table or a probability mass function (pmf).

Two key summary measures:

• Expected value (mean):

$${\mu }_X=E(X)=\sum xP(X=x).$$

• Variance and standard deviation:

$${\sigma }_X^2=\sum (x-{\mu }_X{)}^{2}P(X=x),{\sigma }_X=\sqrt{{\sigma }_X^2}.$$

Expected value is the long-run average, not necessarily a “most likely” outcome. If a game has a negative expected value, it is unfavorable over time even if you sometimes win.

Continuous random variables

A continuous random variable can take on any value in an interval, like height or time. It is described by a density curve. For continuous variables:

• Probabilities are areas under the curve.
• $P(X=a)=0$ for any exact value $a$.
• You focus on intervals, such as $P(a<X<b)$.

The normal distribution is the most important continuous model in AP Statistics, largely because it approximates many natural measurements and plays a major role in sampling distributions.

The Binomial Distribution: Counting Successes

The binomial distribution models the number of successes in $n$ independent trials when each trial has the same probability of success $p$.

To use a binomial model, check the typical conditions:

1. Fixed number of trials $n$
2. Two outcomes per trial (success/failure)
3. Independent trials
4. Constant probability $p$

If $X\sim \text{Bin}(n,p)$, then $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}.$$

The mean and standard deviation are especially useful: $${\mu }_X=np,{\sigma }_X=\sqrt{np(1-p)}.$$

Practical interpretation matters. If $n=100$ and $p=0.02$, then $\mu =2$ means you expect about two successes in the long run, not exactly two every time.

The Normal Distribution: The Workhorse Model

A normal distribution is a symmetric, bell-shaped curve described by mean $\mu$ and standard deviation $\sigma$. If $X\sim N(\mu ,\sigma )$, then standardized values (z-scores) connect any normal distribution to the standard normal: $$z=\frac{x-\mu }{\sigma }.$$

Normal calculations typically involve:

• Converting bounds to z-scores
• Using a table or calculator for areas (probabilities)
• Interpreting the result in context

Normal models are appropriate when the variable is approximately symmetric with a single peak and no extreme outliers. In real data, “approximately normal” is a judgment based on plots and context, not a guarantee.

Combining Random Variables: What Happens When You Add or Subtract?

AP Statistics emphasizes what happens to means and variances when you combine random variables. For $T=aX+bY$:

• The mean transforms linearly:

$${\mu }_T=a{\mu }_X+b{\mu }_Y.$$

For variability, the simplest AP-level results apply when $X$ and $Y$ are independent:

• If $T=X+Y$ or $T=X-Y$, then

$${\sigma }_T^2={\sigma }_X^2+{\sigma }_Y^2.$$

This matters in settings like total cost (sum of independent expenses) or difference in two measurements. Even when the combined mean is easy, the combined standard deviation is where students often slip by adding standard deviations directly. You add variances, not standard deviations.

Sampling Distributions: Distributions of Statistics

A sampling distribution describes how a statistic (like $\bar{x}$ or $\hat{p}$) varies from sample to sample. This is the gateway to inference: confidence intervals and hypothesis tests depend on these distributions.

The sample mean $\bar{x}$

If you repeatedly take samples of size $n$ from a population with mean $\mu$ and standard deviation $\sigma$, then:

• The sampling distribution of $\bar{x}$ has

$${\mu }_{\bar{x}}=\mu ,{\sigma }_{\bar{x}}=\frac{\sigma }{\sqrt{n}}.$$

If the population is normal, then $\bar{x}$ is normal for any $n$. If the population is not normal, the Central Limit Theorem explains that $\bar{x}$ becomes approximately normal for sufficiently large $n$.

A key idea is that increasing $n$ reduces the standard deviation of $\bar{x}$. Larger samples produce more stable averages, which is why “bigger samples mean more precise estimates” is true in a mathematical sense.

The sample proportion $\hat{p}$

For a sample proportion $\hat{p}$ from a population proportion $p$:

• $${\mu }_{\hat{p}}=p,{\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}.$$

The normal approximation for $\hat{p}$ works well when the expected counts of successes and failures are large enough (often stated as $np\ge 10$ and $n(1-p)\ge 10$). This is why very small samples or extremely rare events can break normal-based methods.

Putting It All Together

Probability rules help you compute chances in multi-event situations. Random variables and their distributions turn those chances into models you can summarize with means and standard deviations. Binomial distributions handle counts of successes; normal distributions handle a wide range of measured variables and power many approximations. Combining random variables shows how totals and differences behave. Sampling distributions explain why statistics vary and why larger samples usually lead to more reliable conclusions.

If you can move confidently between context, probability notation, and distribution-based calculations, you have the toolkit AP Statistics expects for probability and distributions, and you are ready for inference.