Multiplication Rule
A method for finding the probability that both of two events occur.
Multiplication Rule: If events A and B are independent, then P(A and B)=P(A)P(B). Otherwise, P(A and B)=P(A)P(B∣A).
Example 1: Suppose a fair six-sided die is rolled twice and that the rolls are independent. Given events
A = rolling a 1 on the first roll
B = rolling a 1 on the second roll
the formula says that P(rolling 1 on both rolls)=P(A and B)=P(A)P(B)=61⋅61=361.
Example 2: A bag contains 3 red balls and 3 black balls. One ball is drawn from the bag, set aside, and then a second ball is drawn. Consider the events
A = the first ball is red
B = the second ball is red.
Note that events A and B are not independent. The color of the first ball has an effect on the probability of the color of the second ball.
Also note that P(A)=63=21 and P(B∣A)=52.
The multiplication rule says that P(A and B)=P(A)P(B∣A)=21⋅52=51. Thus the probability both the first and second balls are red is 51.
See
also
Independent
events, addition
rule, conditional
probability
Worked Example
Problem: A bag contains 5 red marbles and 3 blue marbles. You draw two marbles without replacement. What is the probability that both marbles are red?
Step 1: Identify event A and event B. Let A = first marble is red, and B = second marble is red.
Step 2: Find the probability that the first marble is red. There are 5 red marbles out of 8 total.
P(A)=85 Step 3: Find the conditional probability that the second marble is red, given the first was red. After removing one red marble, 4 red marbles remain out of 7 total.
P(B∣A)=74 Step 4: Apply the multiplication rule by multiplying the two probabilities.
P(A and B)=85⋅74=5620=145 Answer: The probability of drawing two red marbles is 145≈0.357. Another Example
This example uses independent events, so the formula simplifies to P(A) · P(B) without needing conditional probability. The first example involved dependent events (drawing without replacement), where the conditional probability was essential.
Problem: You roll a fair six-sided die and flip a fair coin. What is the probability of rolling a 3 and getting heads?
Step 1: Identify the events. Let A = rolling a 3, and B = getting heads. Since the die roll and coin flip do not affect each other, these events are independent.
Step 2: Find each individual probability.
P(A)=61,P(B)=21 Step 3: Because the events are independent, P(B∣A)=P(B). The multiplication rule simplifies to:P(A and B)=P(A)⋅P(B)=61⋅21=121 Answer: The probability of rolling a 3 and getting heads is 121≈0.083.
Frequently Asked Questions
What is the difference between the multiplication rule for independent and dependent events?
For independent events, the occurrence of one event does not change the probability of the other, so the rule simplifies to P(A and B)=P(A)⋅P(B). For dependent events, the second probability changes based on the first event's outcome, so you must use the full formula P(A and B)=P(A)⋅P(B∣A). The independent version is actually a special case of the general rule, since P(B∣A)=P(B) when events are independent. When do you use the multiplication rule vs the addition rule?
Use the multiplication rule when you want the probability that both events occur (A and B). Use the addition rule when you want the probability that at least one of the events occurs (A or B). A helpful way to remember: "and" means multiply, "or" means add (with an adjustment for overlap).
Can the multiplication rule be extended to more than two events?
Yes. For three events A, B, and C, the rule extends to P(A and B and C)=P(A)⋅P(B∣A)⋅P(C∣A and B). Each successive probability is conditioned on all the previous events having occurred. If all events are independent, this simplifies to P(A)⋅P(B)⋅P(C).
Multiplication Rule vs. Addition Rule
| Multiplication Rule | Addition Rule |
|---|
| Question it answers | What is the probability that A and B both occur? | What is the probability that A or B (or both) occurs? |
| General formula | P(A and B) = P(A) · P(B | A) | P(A or B) = P(A) + P(B) − P(A and B) |
| Simplified case | P(A) · P(B) when events are independent | P(A) + P(B) when events are mutually exclusive |
| Key word clue | "and", "both", "all" | "or", "either", "at least one" |
| Result compared to individual probabilities | Always ≤ the smaller individual probability | Always ≥ the larger individual probability (for positive probabilities) |
Why It Matters
The multiplication rule appears throughout statistics courses, from basic probability problems to more advanced topics like Bayes' theorem and probability distributions. You will use it whenever a problem asks for the likelihood that multiple conditions are all satisfied — for example, passing both a written test and a practical exam, or drawing a specific sequence of cards. Understanding when events are independent versus dependent is a critical skill tested on the AP Statistics exam and standardized tests like the SAT and ACT.
Common Mistakes
Mistake: Multiplying individual probabilities without adjusting for dependence.
Correction: When events are dependent (such as drawing cards without replacement), the probability of the second event changes after the first event occurs. Always check whether you need to use P(B | A) rather than P(B). For example, drawing two aces from a deck without replacement uses P(B | A) = 3/51, not 4/52.
Mistake: Using the multiplication rule when the problem asks for "or" instead of "and".
Correction: The multiplication rule finds P(A and B). If a problem asks for the probability of A or B occurring, you need the addition rule: P(A or B) = P(A) + P(B) − P(A and B). Read the problem carefully for keywords like "both" (multiply) versus "either" (add).
