Unit 2 · Topic 2.7 · about 25 minutes

Independent Events and Unions of Events

Check whether two events are independent, find the probability that both happen when they are, and find the probability that at least one of two events happens.

Predict first

Pick a student at random and ask for their favorite subject. Event A is "the favorite is math" and event B is "the favorite is art". Some students choose each one, and nobody can name two favorites, so A and B are mutually exclusive. Are A and B independent?

Independent events

Events A and B are independent if knowing whether A has occurred does not change the probability that B occurs. In symbols, A and B are independent exactly when

P (B | A) = P (B)

and in that case $P (A | B) = P (A)$ as well. If the equation fails, the events are dependent.

To check with data, compare a conditional probability with the unconditional one. The table below sorts 500 customers of an online store by the device they used and whether they bought something.

500 online store visits by device and purchase (counts)
	Bought something	Bought nothing	Total
Phone	60	240	300
Computer	40	160	200
Total	100	400	500

Choose one of the 500 visits at random. Overall, $P (bought) = \frac{100}{500} = 0.20$ . Given a phone, $P (bought | phone) = \frac{60}{300} = 0.20$ . Given a computer, $\frac{40}{200} = 0.20$ . Learning the device leaves the chance of a purchase exactly where it was, so for these customers "used a phone" and "bought something" are independent.

Real samples rarely line up this neatly, even when two variables have nothing to do with each other in the population. Deciding whether a gap between two percentages is bigger than chance alone would produce is the job of 3.14 Setting Up a Chi-Square Test for Homogeneity or Independence.

Multiplying for independent events

When A and B are independent, $P (B | A)$ is just $P (B)$ , so the general multiplication rule from 2.6 Conditional Probability simplifies to

P (A \cap B) = P (A) \cdot P (B)

A player who makes 80% of her free throws, with each shot independent of the others, makes two in a row with probability $0.80 \cdot 0.80 = 0.64$ . The same equation is a test: if $P (A \cap B) = P (A) \cdot P (B)$ , the events are independent, and if not, they are dependent.

Independence also makes "at least one" questions quick, using the complement from 2.4 Introduction to Probability. A home has three smoke detectors, and each works with probability 0.9, independently of the others. All three fail with probability $0.1 \cdot 0.1 \cdot 0.1 = 0.001$ , so $P (at least one works) = 1 - 0.001 = 0.999$ .

The union: A or B

The union of A and B is the event that A occurs or B occurs or both do. Its probability is written $P (A \cup B)$ . Adding $P (A)$ and $P (B)$ counts every outcome in the overlap twice, so the rule subtracts the overlap once:

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

For mutually exclusive events the overlap is 0, as in 2.5 Mutually Exclusive Events, and the rule becomes plain addition. Notice the word "or" here includes both. "Takes AP Statistics or plays a varsity sport" includes the students who do both.

Worked exampleStatistics students and varsity athletes

At a large high school, 30% of students take AP Statistics, 25% play a varsity sport, and 9% do both. A student is chosen at random. Find (a) the probability that the student takes AP Statistics or plays a varsity sport, (b) the probability that the student does neither, and (c) whether taking AP Statistics and playing a varsity sport are independent.

Name the events. Let S be "takes AP Statistics" and V be "plays a varsity sport". Then $P (S) = 0.30$ , $P (V) = 0.25$ and $P (S \cap V) = 0.09$ .
(a) Use the addition rule. $P (S \cup V) = 0.30 + 0.25 - 0.09 = 0.46$ . The 9% who do both were counted in both of the first two terms, so they are subtracted once.
(b) Use the complement. "Neither" is the complement of "S or V", so the probability is $1 - 0.46 = 0.54$ .
(c) Compare a conditional probability with an unconditional one. $P (S | V) = \frac{0.09}{0.25} = 0.36$ , but $P (S) = 0.30$ . The multiplication test agrees: independence would need $P (S \cap V) = 0.30 (0.25) = 0.075$ , and it is 0.09 instead. The events are not independent. Varsity athletes at this school are more likely than students in general to take AP Statistics.

Answer.

(a) 0.46 (b) 0.54 (c) Not independent, because $P (S | V) = 0.36$ is not equal to $P (S) = 0.30$ .

Two ideas that get confused
	Mutually exclusive	Independent
What it means	A and B cannot happen together	Knowing A happened does not change the probability of B
How to check	$P (A \cap B) = 0$	$P (B \| A) = P (B)$ , or $P (A \cap B) = P (A) \cdot P (B)$

Check your understanding

In a town, 50% of households have a dog, 40% have a cat and 20% have a dog and a cat. For a randomly chosen household, are "has a dog" and "has a cat" independent?

A forecast gives a 30% chance of rain on Saturday, a 40% chance of rain on Sunday, and a 15% chance of rain on both days. What is the probability that it rains on at least one of the two days? Give a decimal.

Your answer

A phone's face scan fails 5% of the time, and each attempt is independent of the others. What is the probability that the face scan works on at least one of 3 attempts?

A survey of 200 adults found that 120 drink coffee every day. Of the 80 adults under 30, 40 drink coffee every day. For an adult chosen at random from the 200, are "under 30" and "drinks coffee daily" independent? Choose the response with the correct answer and a valid reason.

Events A and B are mutually exclusive, with $P (A) = 0.3$ and $P (B) = 0.5$ . Which statement is correct?

Practice

Practice until it is automatic

New numbers every time. Each one is checked the moment you answer, with the full working shown.

Addition rule, complements and conditional probability practice page · Probability from two-way tables practice page

Course alignment, for teachers

AP Statistics topic 2.7, Unit 2: Probability, Random Variables, and Probability Distributions.

Skill 3.C: Calculate and estimate expected counts, percentages, probabilities, and intervals.