1136 LearnersLast updated on June 18th, 2025
Multiplication Rule of Probability
The multiplication rule of probability is a fundamental concept of probability theory. It is the relationship between two or more events that occur together. In this topic, we are going to talk about the multiplication rule of probability and where we use it.
What is the Multiplication Rule of Probability?
When two events A and B occur together, the probability of these two events depends on whether they are independent or dependent events. The multiplication rule is used to find the intersection of two or more events. We denote it as P (A ∩ B), it represents the probability that both events occur together.
If the events are independent, the probability of both events is the product of their individual probabilities and if the events are dependent, then the probability is calculated using conditional probability.
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How to calculate the multiplication rule of probability
When calculating probabilities using the multiplication rule, there are two ways to express it based on the type of event. If the events are independent, the formula we would use is:
P (A ∩ B) = P(A) × P(B)
Here, A and B are two events that are independent of each other, which means that the probability that both of these will occur simultaneously is the product of their respective probabilities.
If the events are dependent on each other, it means that the outcome of one event directly affects the outcome of the second event. We would calculate the probability using conditional probability. The formula is:
P (A ∩ B) = P(A) × P(B|A)
P(A) is the probability of event A occurring, and P(B|A) is the conditional probability of event B occurring given that event A has already occurred.
Real-life applications on Multiplication Rule of Probability
There are many uses of the multiplication rule of probability. Let us now see the uses and applications of the multiplication rule in different fields:
- Healthcare: Hospitals and clinics use the multiplication rule to calculate the probability of a patient having any particular disease based on multiple results.
- Weather forecast: To predict the likelihood of complex weather events, meteorologists use the multiplication rule of probability. This can help show whether there is going to be high humidity due to a storm occurring.
- Marketing campaigns: The multiplication rule helps estimate the overall success rate of a campaign by taking into consideration the probabilities of multiple independent factors, such as audience engagement, advertisements, etc.
Common Mistakes and How to Avoid Them in Multiplication Rule of Probability
Students might make mistakes when learning about the multiplication rule of probability. So here are a few mistakes that students make and ways to avoid them:
Mistake 1
Confusing dependent events with independent events
Students assume all events are independent and end up using the formula P(A ∩ B) = P(A) × P(B). First, we need to check whether the second event depends on the first. If it does, it is a dependent event, and we need to use the conditional probability:
P(A∩B) = P(A) × P(B∣A).
Mistake 2
Not correctly interpreting the intersection symbol
Students might get the symbol for an intersection confused. They may think that P(A∩B) is P(A) + P(B) instead of using the multiplication of probabilities. Remember that ∩ represents “AND” both events occurring together, which is the multiplication of both probabilities.
Mistake 3
Multiplying probabilities incorrectly
Make sure to read the problems before trying to multiply the probabilities. Sometimes, students may try to multiply probabilities where the events are mutually exclusive. This cannot happen because their intersection is 0.
Mistake 4
Not converting the percentages into probabilities
When calculating probabilities, students should remember to convert the percentage into probabilities. If it is not converted, then the answer we get will be incorrect.
Mistake 5
Forgetting that probability must be between 0 and 1.
When doing calculations, the probability must be between 0 and 1. If the answers are greater than 1 or negative, then there is a calculation error. So make sure the answer comes between 0 and 1.
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Solved examples on Multiplication Rule of Probability
Problem 1
A deck has 52 cards. What is the probability of drawing two aces in a row without replacement?
0.0045
Explanation
P(Ace1 ∩ Ace2) = 452 × 351
= 122652
= 0.0045
Step 1: Probability of drawing the first ace = 4 / 52
Step 2: Since one ace has been removed, the probability of drawing a second ace = 3 / 51 .
Step 3: Multiply both probabilities: 4 / 52 × 3 / 51
Problem 2
What is the probability of getting two heads when flipping two fair coins?
1/4
Explanation
P(H1 ∩ H2) = 1 / 2 × 1 / 2 = 1 / 4
Step 1: Probability of getting heads on the first flip = 1 / 2
Step 2: Probability of getting heads on the second flip = 1 / 2
Step 3: Multiply both: 1 / 2 × 1 / 2
Problem 3
A bag has 5 red and 10 blue marbles. If you pick two marbles with replacement, what is the probability of getting two red ones?
1/9
Explanation
P(R1 ∩ R2) = 5 / 15 × 5 / 15 = 1 / 9
Step 1: Probability of first red = 5 / 15
Step 2: Since replacement occurs, second red = 5 / 15.
Step 3: Multiply 5 / 15 × 5 / 15
Problem 4
A group has 6 females and 4 males. What is the probability of randomly selecting two females?
1/3
Explanation
P(F1 ∩ F2) = 6/10 × 5/9 = 30/90 =1/3
Step 1: Probability of first female = 6/10
Step 2: Probability of the second female (after one is removed) = 5/9
Step 3: Multiply: 6/10 × 5/9
Problem 5
Machine A has a failure probability of 0.1, and Machine B has 0.2. What is the probability that both fail?
0.02
Explanation
P(A ∩ B) = 0.1 × 0.2 = 0.02
Step 1: Probability of failure of A = 0.1
Step 2: Probability of failure of B = 0.2
Step 3: Multiply: 0.1 × 0.2
Problem 6
A factory makes 5% defective items. What is the probability of picking two defective ones?
0.0025
Explanation
P(D1 ∩ D2) = 0.05 × 0.05 = 0.0025
Step 1: Probability of first defective = 0.05
Step 2: Probability of second defective = 0.05
Step 3: Multiply: 0.05 × 0 .05
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FAQs on Multiplication Rule of Probability
1.What is the multiplication rule of probability?
2.How can we determine that two events are independent?
3.Can the multiplication rule be used for more than two events?
4.What happens if both events A and B are mutually exclusive?
5.Can the probability of the events be greater than 1?
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
