AP Statistics · Topic 2.7 · Unit 2

AP Stats 2.7: Independent Events & Unions

By Jude Wallis · Published July 28, 2026

Topic 2.7 in the Fall 2026 AP Statistics course covers independence and unions. Events are independent when one occurring does not change the probability of the other, so P(A and B) equals P(A) times P(B). The union rule is P(A or B) equals P(A) plus P(B) minus P(A and B).

AP Statistics: Unit 2 (topics 2.7). In the Fall 2026 AP Statistics course, independent events and unions of events are Unit 2 Topic 2.7, aligned to skill 3.C.

What topic 2.7 covers

Topic 2.7 handles two related calculations: probabilities for independent events and probabilities for the union of two events. The objective is to calculate both.

These build directly on the conditional probability from topic 2.6. Independence is the special case where conditioning changes nothing, and the union rule is the 'or' companion to the multiplication rule for 'and'.

Independent events

Events $A$ and $B$ are independent if and only if knowing whether $A$ has occurred does not change the probability that $B$ will occur. Equivalently, $P(A \mid B) = P(A)$ and $P(B \mid A) = P(B)$ .

When that holds, the general multiplication rule simplifies, because the conditional probability collapses to an unconditional one: $P(A \cap B) = P(A) \cdot P(B)$ . You can use this in two directions: multiply the probabilities when you already know the events are independent, or check independence by testing whether $P(A \cap B)$ actually equals $P(A) \cdot P(B)$ . Independence is not the same as being disjoint, a distinction drawn out in disjoint vs independent events.

The union of two events

The probability that event $A$ or event $B$ (or both) occurs is the probability of the union, written $P(A \cup B)$ . The general addition rule is:

P(A \cup B) = P(A) + P(B) - P(A \cap B)

You add the two individual probabilities, then subtract the joint probability so the overlap is not counted twice. If the events are mutually exclusive from topic 2.5, the intersection is 0 and the rule reduces to $P(A) + P(B)$ . The subtraction is the only part students routinely drop, so identify $P(A \cap B)$ before you add.

Common mistakes with independence and unions

Two errors recur on these problems. The first is dropping the subtraction in the addition rule, which double-counts the outcomes that satisfy both events; always subtract the joint probability unless the events are disjoint.

The second is assuming independence without checking it, then multiplying probabilities that should not be multiplied. Test independence directly by comparing the product $P(A) \cdot P(B)$ with the joint probability $P(A \cap B)$ , or by comparing a conditional probability with its unconditional version. Sketching the sample space or a two-way table keeps both traps visible, and it reminds you that the multiplication shortcut only applies once independence is confirmed.

Red cards, face cards, independence, and a union

Draw one card from a standard 52-card deck. Let A be 'red card' (26 cards) and B be 'face card' (12 cards). There are 6 red face cards. Check whether A and B are independent, then find P(A or B).

Individual probabilities: $P(A) = 26/52 = 0.5$ and $P(B) = 12/52 = 0.2308$ .
Joint probability of a red face card: $P(A \cap B) = 6/52 = 0.1154$ .
Independence check: $P(A) \cdot P(B) = 0.5 \times 0.2308 = 0.1154$ , which equals $P(A \cap B)$ , so the events are independent.
Union rule: $P(A \cup B) = 26/52 + 12/52 - 6/52 = 32/52 = 0.6154$ .

The events are independent because $P(A \cap B) = P(A)\cdot P(B) = 0.1154$ , and $P(A \text{ or } B) = \tfrac{32}{52} \approx 0.615$ .

Frequently asked questions

How do you check if two events are independent?

Test whether P(A and B) equals P(A) times P(B). If the joint probability matches the product of the two individual probabilities, the events are independent. Equivalently, they are independent when P(A given B) equals P(A), meaning the condition does not change the probability.

What is the general addition rule?

It gives the probability of a union: P(A or B) equals P(A) plus P(B) minus P(A and B). You subtract the joint probability so outcomes in both events are not counted twice. If the events are mutually exclusive, that overlap is 0 and the rule becomes P(A) plus P(B).