2.4 Find intersections using the multiplication rule | Athabasca University MATH 215: Introduction to Statistics

Athabasca University MATH 215: Introduction to Statistics

Probability

Combine probabilities to find the chance that events happen together

Sometimes a question asks for the chance that two events both occur. For example, a student might ask for the chance that two items selected in sequence are both red. The word “both” signals an intersection: the outcome must belong to each event at the same time. The multiplication rule finds this probability by combining the chance of the first event with the chance of the second event after the first has occurred. The second chance may change, so it is important to decide whether the events are independent before using the shorter form of the rule.

What you will learn

Explain what an intersection of events means.
Use the multiplication rule with a conditional probability.
Use the independent-events form of the multiplication rule when independence is justified.
Interpret a calculated intersection probability in context.

Events and intersections

An experiment is an activity with an outcome that is not known in advance, such as selecting an item at random. An event is a set of outcomes that meet a stated condition. For example, when selecting a token, event

A

could mean “the token is red.” The probability of an event, written

P(A)

, is a number from 0 to 1 that describes how likely the event is.

The intersection of two events is the event that both conditions are true. The notation

A \cap B

means “

A

and

B

.” Thus,

P(A \cap B)

is the probability that both events occur in the same experiment. This is different from asking for the probability that at least one event occurs.

The multiplication rule connects an intersection probability to a conditional probability. A conditional probability is a probability calculated with additional information already known. The notation

P(B \mid A)

means “the probability of

B

, given that

A

has occurred.” The vertical bar does not mean division; it marks the condition.

“Both” usually signals an intersection.
$P(B \mid A)$ describes the chance of $B$ after event $A$ is known to have occurred.
Keep track of which event is being treated as the condition.

The multiplication rule

To use the general multiplication rule, describe the two events clearly and identify the order or information being used. The rule is valid when the relevant probabilities are defined and the conditional probability is meaningful. In particular, the probability of the conditioning event

A

must be greater than zero.

First find the probability of

A

. Then find the probability of

B

under the condition that

A

has occurred. Multiply these two values to get the probability that both happen. This is ordinary algebra: multiplication combines the two factors, just as in a product such as

3 \times 4

. Here, the factors represent probabilities, so the result is also a probability.

If the first event does not change the probability of the second, the events are independent. Independence means that knowing

A

occurred does not change the chance of

B

. In that case, the conditional probability equals the original probability of

B

, and the multiplication rule can be written in its shorter form. Do not assume independence merely because two events have different names; check what information about the first event does to the second probability.

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

Use a conditional probability when the chance of the second event depends on the first.
Use the shorter independent-events form only when independence is justified.
The intersection probability cannot be larger than either of its component event probabilities.

Decide whether the second probability changes

A common setting is sampling without replacement: once an item is selected, it is not put back before the next selection. The total number of available items changes, and the number of items of a particular kind may change too. As a result, the probability on the second selection may depend on what happened on the first selection.

With replacement, the selected item is returned before the next selection. If the selections are carried out in the same way and returning the item leaves the relevant chances unchanged, the events may be independent. The important check is not the label “with replacement” by itself, but whether knowing the first result changes the probability of the second event.

Write the event descriptions before substituting numbers. For example, if

A

means “the first item is red” and

B

means “the second item is red,” then the requested probability that both are red is

P(A \cap B)

. The conditional factor is specifically the probability that the second item is red given that the first was red. This wording helps prevent mixing up the first and second selection.

Check whether the first result affects the second probability.
For a sequence, keep the event order clear.
A conditional probability must match the condition stated in the question.

Calculate and communicate the result

After finding the two factors, multiply them and state what the answer means in the original setting. A probability may be reported as a decimal, a fraction, or a percentage. Keep enough digits during the calculation to avoid rounding error, then round the final result as requested.

A decimal probability can be interpreted as a long-run proportion: if the same kind of experiment were repeated many times under the same conditions, the proportion of trials with both events would tend to be near that probability. This interpretation describes likelihood, not a guarantee about any one trial.

Before accepting an answer, check that it is between 0 and 1 and no greater than either event probability. If it is larger than one of them, the result cannot represent the chance that both events occur, because both happening is more restrictive than either one happening alone.

Multiply the appropriate probabilities, then interpret the result in context.
Round only the final answer.
Check that the final probability is between 0 and 1.

Worked example

Two red tokens selected without replacement

A bag contains 5 red tokens and 3 blue tokens. Two tokens are selected one after the other without replacement. Find the probability that both selected tokens are red.

Name the events
Let $A$ be the event that the first token is red, and let $B$ be the event that the second token is red. The request is for the intersection $A \cap B$ , because both conditions must occur. There are 8 tokens initially: 5 red and 3 blue.
Check the conditions
The selection procedure gives a defined probability for each event. Since the first token is not returned, learning that the first token was red changes the contents of the bag: 4 red tokens remain among 7 total tokens. Therefore, use the general multiplication rule and a conditional probability rather than assuming independence.
Write the rule
The probability that the first token is red is 5 out of 8. Given a red first token, the probability that the second is red is 4 out of 7. Substitute these values into the multiplication rule.
$P(A \cap B)=P(A)P(B \mid A)=\frac{5}{8}\times\frac{4}{7}$
Calculate
Multiply the numerators and denominators. The product is 20 out of 56, which simplifies to 5 out of 14. As a decimal, this is approximately 0.357142; round the final probability to four decimal places.
$\frac{5}{8}\times\frac{4}{7}=\frac{20}{56}=\frac{5}{14}\approx 0.3571$

Answer: The probability that both selected tokens are red is

\frac{5}{14}

, or approximately 0.3571 (35.71%).

Check: The result is between 0 and 1 and is less than the probability of a red first token,

5/8

. Both checks are consistent with an intersection probability.

Common mistakes and how to avoid them

Multiplying the two original probabilities even though the first event changes the second probability.

Correction: Use the conditional probability for the second factor. In the token example, after a red first token, 4 of the 7 remaining tokens are red.

Treating

P(B \mid A)

as the probability of

A

and

B

together.

Correction:

P(B \mid A)

is the chance of

B

under the condition that

A

occurred. Multiply it by

P(A)

to find the intersection.

Assuming that the multiplication rule applies only to independent events.

Correction: The general rule uses a conditional probability and works whether or not the events are independent. Independence only allows the conditional factor to be replaced by

P(B)

.

Reporting a probability greater than 1 or greater than either event probability.

Correction: Recheck the factors and multiplication. An intersection is no more likely than either event considered alone.

Lesson summary

An intersection, $A \cap B$ , means that both events occur.
The general multiplication rule is $P(A \cap B)=P(A)P(B \mid A)$ .
When $A$ and $B$ are independent, use $P(A \cap B)=P(A)P(B)$ .
Check whether the first event changes the probability of the second, calculate carefully, and interpret the result in context.

Check your understanding

Question 1

A jar has 4 green and 6 yellow beads. One bead is selected and kept out of the jar; then a second is selected. Let

A

mean the first bead is green and

B

mean the second bead is green. What is

P(A \cap B)

?

$\frac{4}{10}\times\frac{3}{9}=\frac{2}{15}$
$\frac{4}{10}\times\frac{4}{10}=\frac{4}{25}$
$\frac{4}{10}+\frac{3}{9}$
$\frac{3}{9}$

Show answer and explanation

\frac{4}{10}\times\frac{3}{9}=\frac{2}{15}

Given a green first bead, 3 green beads remain among 9 total beads. Thus the intersection probability is

\frac{4}{10}\times\frac{3}{9}=\frac{12}{90}=\frac{2}{15}

.

Question 2

Suppose two events are independent, with

P(A)=0.30

and

P(B)=0.40

. What is the probability that both occur?

0.12
0.70
0.30
0.40

Show answer and explanation

0.12

Because the events are independent, multiply their probabilities:

0.30\times0.40=0.12

.

Question 3

Which expression gives the general multiplication rule for the intersection of

A

and

B

?

$P(A)P(B \mid A)$
$P(A)+P(B)$
$P(A)-P(B)$
$P(B \mid A)$

Show answer and explanation

P(A)P(B \mid A)

The general rule multiplies the chance of

A

by the chance of

B

given that

A

occurred.

Key terms

Event: A set of outcomes that meet a stated condition.
Intersection: The event that two specified events both occur, written $A \cap B$ .
Conditional probability: The probability of an event when another event is known to have occurred; $P(B \mid A)$ means the probability of $B$ given $A$ .
Independent events: Events for which knowing that one occurred does not change the probability of the other.
Multiplication rule: A rule for finding an intersection probability by multiplying one event's probability by a suitable conditional probability.

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Published by DoAssignment. This AI-assisted lesson follows Athabasca University MATH 215: Introduction to Statistics, study topic 2.4. It is a study resource, not an official curriculum publication.

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2.4 · Find intersections using the multiplication rule

Athabasca University MATH 215: Introduction to Statistics

Combine probabilities to find the chance that events happen together

What you will learn

Events and intersections

The multiplication rule

Decide whether the second probability changes

Calculate and communicate the result

Two red tokens selected without replacement

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MATH 215

About this lesson