Math Core

Sometimes learning that one event happened tells you nothing at all about another. A coin doesn't remember the last flip, and one randomly chosen customer's purchase doesn't affect the next one's. When events don't influence each other's probabilities, they are independent, and probability calculations become much simpler.

What independence means

Definition

Independent events

Events $A$ and $B$ are independent if knowing whether one occurs does not change the probability of the other:

$P(A \mid B) = P(A).$

Equivalently, $P(B \mid A) = P(B)$ , or $P(A \text{ and } B) = P(A) \cdot P(B)$ .

Any one of those three equations is enough to show independence, and if any one fails, the events are dependent.

Checking independence in a two-way table

Compare a conditional probability with the overall (unconditional) probability.

Worked example: Are grade and phone type independent?

A survey of 220 students recorded grade level and phone type.

	iPhone	Android	Total
9th grade	38	22	60
10th grade	45	35	80
11th grade	52	28	80
Total	135	85	220

Are the events "Android" and "9th grade" independent for a randomly chosen student?

Solution. Compare

$P(\text{Android}) = \dfrac{85}{220} \approx 0.386 \quad\text{with}\quad P(\text{Android} \mid \text{9th}) = \dfrac{22}{60} \approx 0.367.$

These are not equal, so the events are not independent. Knowing a student is a ninth grader slightly lowers the chance the student uses Android.

In real data, conditional and overall proportions are rarely exactly equal. For this lesson, "independent" means exactly equal. (Later in the course, a chi-square test will tell you whether a difference is big enough to be convincing.)

Worked example: Checking with the multiplication rule

Suppose $P(A) = 0.4$ , $P(B) = 0.25$ and $P(A \text{ and } B) = 0.1$ . Are $A$ and $B$ independent?

Solution. $P(A) \cdot P(B) = 0.4 \times 0.25 = 0.1$ , which equals $P(A \text{ and } B)$ . So $A$ and $B$ are independent. You could also check $P(A \mid B) = \dfrac{0.1}{0.25} = 0.4 = P(A)$ .

The multiplication rule for independent events

When events are independent, $P(B \mid A)$ is just $P(B)$ , so the general multiplication rule simplifies:

Multiplication rule for independent events

If $A$ and $B$ are independent,

$P(A \text{ and } B) = P(A) \cdot P(B).$

This extends to any number of independent events: multiply their probabilities.

Worked example: Connecting flights

Your first flight is on time with probability $0.85$ , and your connecting flight is on time with probability $0.90$ . Assuming the flights are independent, what is the probability both are on time?

Solution. $P(\text{both on time}) = 0.85 \times 0.90 = 0.765$ .

Is independence reasonable here? Maybe not: bad weather at a hub could delay both. Always think about whether the assumption makes sense in context.

"At least one" problems

"At least one" means one, two, three, … up to all of them, which is many cases. The complement, "none", is a single case, and with independence it's easy to compute.

$P(\text{at least one}) = 1 - P(\text{none}).$

Worked example: At least one failure

A machine has 4 components that work independently. Each component fails during a year with probability $0.1$ . What is the probability that at least one component fails during the year?

Solution. Each component survives with probability $0.9$ , so

$P(\text{none fail}) = 0.9^4 = 0.6561, \qquad P(\text{at least one fails}) = 1 - 0.6561 = 0.3439.$

Independent is not the same as mutually exclusive

These two ideas are easy to confuse, but they are nearly opposites. Mutually exclusive events can't happen together, so if you know $A$ happened, $B$ definitely did not. That is very strong information about $B$ . So two mutually exclusive events with positive probabilities are always dependent.

Common mistake

Don't multiply probabilities unless the events are independent, and don't call events independent just because they are "different" or "separate". Check with $P(A \mid B) = P(A)$ or $P(A \text{ and } B) = P(A)P(B)$ , or justify independence from how the data were produced (for example, separate flips of a coin or a random sample from a large population).

Practice

Practice 1

Events $A$ and $B$ are independent with $P(A) = 0.3$ and $P(B) = 0.5$ . Find $P(A \text{ and } B)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

For the events in the previous problem, find $P(A \text{ or } B)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

For events $A$ and $B$ , $P(A) = 0.6$ , $P(B) = 0.5$ and $P(A \text{ and } B) = 0.3$ . Which statement is true?

Practice 4

A random sample of 400 adults were asked whether they exercise regularly.

	Yes	No	Total
Under 40	128	72	200
40 and over	96	104	200
Total	224	176	400

Are the events "under 40" and "exercises regularly" independent?

Practice 5

A salesperson makes 5 independent calls, and each call results in a sale with probability $0.2$ . Find the probability of at least one sale. Round to 4 decimal places.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Events $A$ and $B$ are mutually exclusive, with $P(A) = 0.3$ and $P(B) = 0.4$ . Which statement is true?

Practice 7

Events $A$ and $B$ are independent, $P(A) = 0.4$ and $P(A \text{ or } B) = 0.7$ . Find $P(B)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.