Multi-event Probability: Multiplication Rule

← The Monty Hall Problem

Probability is an incredibly important part of statistics! We can use probability to help us make decisions under uncertainty and make predictions.

Probability (or chance) is the percentage of times one expects a certain outcome when the process is repeated over and over again under the same conditions.

The chance of an outcome is equal to: the # of that outcome / total # of possibilities

Probability Notation

Oftentimes, data scientists use probability notation to express different probabilities:

  • Example: P(A) is read as “the probability of event A”

We can calculate simple probabilities with games of chance. For example...

  • The probability of rolling a fair 6 sided die and getting a 1, is 1/6 because there is only 1 side marked "1" among 6 total sides.

  • The probability of drawing the queen of hearts from a standard deck of 52 cards is 1/52 because there is only one queen of hearts among 52 total cards.

Probabilities can be expressed as fractions (shown above), decimals, or percentages.

Multiplication Rule (Independent Events)

Sometimes, we may want to look at more complicated probabilities, such as the probability that two things happen at the same time. To do this, we can use The Multiplication Rule.

When we have two independent events, the Multiplication Rule is:

P(A and B) = P(A) × P(B)
When A and B are independent events

Example: Rolling Two Dice

The probability of rolling two dice and getting one marked "1" and one marked "2"" can be found using the Multiplication Rule:

Multiplication Rule (Dependent Events)

For dependent events, the multiplication rule is

P(A and B) = P(A) * P(B | A), where P(B | A) is the probability of event B given that event A happened.

P(A and B) = P(A) × P(B | A)
When P(B | A) is the probability of event B given that event A happened.

Example: Drawing Two Cards without Replacement

The probability of drawing 2 cards without replacement from a deck and getting a heart and then a spade can be found using the Multiplication Rule.

Example: Flipping Four Coins

The multiplication rule can be used for more than 2 events as well. Here's a simple example with flipping a fair coin.

For example, what’s the probability of flipping a fair coin 4 times and getting all heads?

Example Walk-Throughs with Worksheets

Video 1: Multiplication Rule Examples

Follow along with the worksheet to work through the problem:

Video 2: Multiplication Rule Special Cases

Follow along with the worksheet to work through the problem:

Practice Questions

Q1: When you read a word problem about probability, the word ‘and’ signals that you should use the multiplication rule.
Q2: Which of the following is correct when describing a full, normal deck of cards?
Q3: What is the probability of drawing two cards **with replacement** from a deck and getting a spade and a club
Q4: What is the probability of drawing two cards **without replacement** from a deck and getting a heart and a diamond?
Q5: What is the probability of rolling three dice and getting the first one marked as 1, the second one marked as 2, and the third one marked as three?
← The Monty Hall Problem