The multiplication rule in probability allows you to calculate the probability of multiple events occurring together using known probabilities of those events individually. There are two forms of this rule, the specific and general multiplication rules.

In this post, learn about when and how to use both the specific and general multiplication rules. Additionally, I’ll use and explain the standard notation for probabilities throughout, helping you learn how to interpret it. We’ll work through several example problems so you can see them in action. There’s even a bonus problem at the end!

Before we get to the rules themselves, you need to know the definitions for independent and dependent events:

**Independent events**: The occurrence of one event does not affect the probability of the other event. For example, when flipping a coin, getting “heads” does not change the likelihood of getting “heads” on the next coin flip.**Dependent events**: The occurrence of one event does affect the probability of the other event. For example, if you draw a King from a deck of cards and do not replace it, it causes the probability of drawing another King to decrease.

Using the multiplication rule, you can calculate the probability that events A and B occur jointly when you know the probability of event A and event B occurring individually.

The notation for the joint probability of A and B occurring is the following: P(A ∩ B).

When events are independent, you can use the specific multiplication rule. When you have dependent events, you must use the general multiplication rule.

**Related post**: Probability Fundamentals

## Specific Multiplication Rule

Use the specific multiplication rule to calculate the joint probability of independent events. To use this rule, multiply the probabilities for the independent events. With independent events, the occurrence of event A does not affect the likelihood of event B. This rule is not valid for dependent events.

Using probability notation, the specific multiplication rule is the following:

P(A ∩ B) = P(A) * P(B)

Or, the joint probability of A and B occurring equals the probability of A occurring multiplied by the probability of B occurring.

### Examples of the Specific Multiplication Rule

For example, to calculate the probability of obtaining “heads” during two consecutive coin flips, multiply the probability of heads on the first coin flip (0.5) by the probability of heads on the second coin flip (0.5).

0.5 X 0.5 = 0.25

The joint probability of two consecutive heads is 0.25.

Imagine you particularly like wearing tan pants with a blue shirt. However, in the morning, you’re sleepy and grab your pants and shirt randomly from the closet. The pants are on one side of the closet while the shirts are on the other. These are independent events because grabbing a pair of pants doesn’t affect the probabilities for shirts.

You have ten pairs of pants and three are tan. Consequently, the probability of drawing a tan pair (event TP) is 0.3.

You have 16 shirts and four are blue. Hence, the probability of grabbing a blue shirt (event BS) is 0.25.

These are independent events because selecting a pair of pants doesn’t affect the likelihood of drawing a blue shirt and vice versa.

Using the specific multiplication rule for these independent events:

P(TP ∩ BS)= P(TP) * P(BS)

0.3 X 0.25 = 0.075

Or, the joint probability of randomly selecting a pair of tan pants and a blue shirt equals 0.075, which is the probability of tan pants multiplied by the probability of a blue shirt.

The likelihood of getting your preferred combination is low! You might want to drink some coffee to increase your chances!

## General Multiplication Rule

Use the general multiplication rule to calculate joint probabilities for either independent or dependent events. When you have dependent events, you must use the general multiplication rule because it allows you to factor in how the occurrence of event A affects the likelihood of event B.

Using standard notation, the general multiplication rule is the following:

P(A ∩ B) = P(A) * P(B|A)

Or, the joint probability of A and B occurring equals the probability of A occurring multiplied by the conditional probability of B occurring given that A occurred.

The difference between the general and specific rules is, unsurprisingly, that you can use the general rule more generally. It works for both independent and dependent events, whereas the specific rule is valid only for independent events.

Why can you use the general form for both independent and dependent events? In the notation, focus on P(B|A), which is the conditional probability that event B occurs given that event A occurred.

In the context of independent events, P(B|A) = P(B) because event A occurring does not impact event B’s probability. That’s the very definition of independent events. Consequently, this rule becomes equivalent to the specific multiplicative rule for independent events.

However, for dependent events, P(B|A) ≠ P(B). That’s just another way of saying that event A occurring affects the probability of event B (i.e., they’re dependent events). The general multiplicative rule allows you to factor in the other event, as you will see in the next two examples!

**Related post**: Using Contingency Tables to Calculate Probabilities

### Examples of the General Multiplication Rule

The classic example for dependent events is drawing cards from a deck of cards without replacement. As you draw cards, it affects the probability of the next card you can draw.

Suppose you are interested in the probability of drawing hearts on two consecutive draws. Initially, the deck has 13 hearts out of its 52 cards (13/52 = 0.25). If you draw a heart (event H1), that changes the probability of drawing another heart. The dependent probability of drawing that second heart (event H2) is now 12/51 = 0.235.

In notation form:

P(H1 ∩ H2) = P(H1) * P(H2|H1)

Or, the joint probability of drawing two consecutive hearts equals the probability of the first heart multiplied by the probability of the second heart given that the first card was a heart.

0.25 * 0.235 = 0.059

### Dependent Events: Tan Pants and Blue Shirts Example

Let’s go back to the pants and shirt example. Imagine that we’re packing for a short trip and randomly select two pairs of pants and two pairs of shirts to include in our suitcase. We’re hoping for two pairs of tan pants and two blue shirts.

We’ll start by treating this as two sets of dependent events, one for pants and the other for shirts.

We begin with 10 pairs of pants, three of which are tan. Consequently, the probability of the first pair of pants being tan (event T1) is 0.30. The probability of the second pair being tan (T2) is 2/9 = 0.22. Hence:

P(T1 ∩ T2) = P(T1) * P(T2|T1)

0.30 * 0.22 = 0.066

The joint probability of drawing two pairs of tan pants is 0.066, which equals the probability of the first pair of tan pants multiplied by the conditional probability of the second pair of tan pants given that the first pair was tan.

And, for the shirts, we start with four blue shirts out of 16 total shirts. Using the same approach, we get the following:

P(B1 ∩ B2) = P(B1) * P(B2|B1)

0.25 * 0.20 = 0.05

We have the two joint probabilities of 0.066 for two tan pants and 0.05 for two blue shirts.

**Related post**: Using Permutations to Calculate Probabilities and Using Combinations to Calculate Probabilities

## Bonus Example Problem!

Our ultimate goal as a random packer is that we’d like to have two tan pants and two blue shirts in our suitcase. Can you figure out how to calculate that probability given the above information?

To solve that problem, we’ll define two tan pants as event 2TP and two blue shirts as event 2BS. From our previous calculations for dependent events using the general multiplication rule, we know the following:

P(2TP) = 0.066

P(2BS) = 0.05

How do you calculate the joint probability P(2TP ∩ 2BS)?

Think back to the example of independent events where we drew one pair of pants and one shirt. Selecting pants does not affect the probabilities for shirts and vice versa. Consequently, we can treat events 2TP and 2BS as independent events even though we had dependent events when calculating probabilities for multiple pants and multiple shirts. In other words, selecting multiple pants affects the likelihood of the next pair of pants, but it does not affect shirts.

Hence, we can use the specific multiplication rule for independent events for this part of the solution:

P(2TP ∩ 2BS) = P(2TP) * P(2BS)

0.066 * 0.05 = 0.0033

The probability of drawing two pairs of tan pants and two blue shirts is only 0.0033 or 0.33%! That’s not too likely to occur by chance. If we genuinely want that combination, we should consider a non-random approach to packing!

Calculating joint probabilities using the multiplication rule is simple. Determine whether your events are independent or dependent, and then use the correct form of the rule!

Steph says

I tried to find an answer to this little open question myself. My conclusion is that the “|” (given) in conditional probabilities, means that you evaluate something after something else has happened but the first thing that happened does not imply (is not a trigger that means something will happen but just an initial situation). The evaluation of a variable will depend on an initial situation but is not triggered by that initial situation.

Steph says

Hi Jim,

It is easy to understand that P(X,Y) = P(Y,X), so the joint probability is commutative.

It is also easy to understand that for independent variables, P(X,Y) = P(X) * P(Y)

Concerning dependent variables, however, I need a clarification.

Mathematically, to get Bayes’ theorem,

we consider :

P(X,Y) = P(X) * P(Y|X)

P(Y,X) = P(Y) * P(X|Y)

to arrive at P(X) * P(Y|X) = P(Y) * P(X|Y) and thus the Bayes’ theorem :

P(Y|X) = P(X|Y) * P(Y) / P(X)

“Dependent” does not only mean correlation but causation.

Let’s imagine X = The sun is shining & Y = ice cream sales.

Causality, by definition, is unidirectional (or else it is a correlation and the term “Dependent” is not correct):

P(X,Y) = P(X) * P(Y|X) is correct in expressing our real situation: The probability of having sunshine multiplied by the probability of selling ice cream given sunshine.

On the other hand, considering P(Y,X) = P(Y) * P(X|Y) should not apply because the probability of having sunshine given the sale of ice cream does not make sense in our situation. Just because I eat ice cream doesn’t mean it will be sunny.

Baye’s theorem considers that causality can go in both directions and that each variable can depend on the other? So it’s more like correlated variables than a dependent variable?

Thank you in advance for your clarification!

Dheerendra says

Thanks Jim, wonderful article.

Yash Mopur says

Hello Jim,

This is a wonderful article. Until now I had a hard time understanding the basics of probability. This is truly enlightening. You have a great way of explaining things. I had read your Introduction to Statistics book and in the process of reading Linear Regression Both these books are gems.

Thanks for everything!

Jim Frost says

Hi Yash,

Thanks so much for your kind words! I’m so glad this blog post and my books have been helpful! ðŸ™‚

Joseph Walkenhorst says

Hi Jim,

A great post breaking down dependent vs independent events and joint probabilities in a straight-forward way. These concepts are critical to interpreting and modelling business outcomes, but are often not applied correctly because of a lack of awareness and understanding.

Great examples too, especially the final example that combines both dependent and independent events together to calculate the joint probability.

BTW I also enjoyed reading your hypothesis testing book, some really intuitive and well written explanations and a well thought out structure.

Cheers,

Joe

Jim Frost says

Hi Joe,

Thanks so much for your kind words. You made my day!

Also, I’m so glad that my hypothesis testing book was helpful! ðŸ™‚

drandywett says

Another well explained post – Thankyou. Is that in one of you books? – I have Intro to stats and Hypothesis testing

Jim Frost says

Thanks! It’s not currently in one of my book. However, I will be writing a probability book down the road!

Thanks for supporting my books too! ðŸ™‚