Code360 powered by Coding Ninjas X Naukri.com. Code360 powered by Coding Ninjas X Naukri.com
Table of contents
1.
Introduction
2.
Theorem
3.
Proof
4.
Examples
5.
FAQs
6.
Key Takeaways
Last Updated: Mar 27, 2024

Multiplication Theorem

Author Akash Nagpal
1 upvote
Master Python: Predicting weather forecasts
Speaker
Ashwin Goyal
Product Manager @

Introduction

Multiplication Theorem derives a relationship between two events. Considering two events, let’s say A and B are associated within a Sample space S. Therefore AB denotes the events in which both events have simultaneously occurred. This theorem is also known as the Multiplication Rule of Probability. So, this is the core idea behind the probability multiplication theorem. Let's get to know it a little better.

Theorem

Considering A and B as two independent events, the probability of both occurring simultaneously would be equal to the product of their individual probabilities.

  • For independent events,
    P(A∩B) = P(A) × P(B)
  • For dependent events, 

P(A ∩ B) = P(A) × P(B|A)

Get the tech career you deserve, faster!
Connect with our expert counsellors to understand how to hack your way to success
User rating 4.7/5
1:1 doubt support
95% placement record
Akash Pal
Senior Software Engineer
326% Hike After Job Bootcamp
Himanshu Gusain
Programmer Analyst
32 LPA After Job Bootcamp
After Job
Bootcamp

Proof

Let us assume that, 

Event A can happen in n1 ways out of which p are successful events 

Event B can happen in n2 ways out of which q are successful events 

Therefore, the number of successful cases = combining the successful events of p and q

Thus, Number of successful cases = p × q

Total cases = n1 × n2 

Therefore, 

P(A and B) = P(A ∩ B) = p × q ⁄ n1 × n2 = p×n1 ⁄ q×n2 

So we are having , P(A) = p ⁄ n1   and P(B) = q ⁄ n2

Hence , P(A ∩ B) = P(A) × P(B)


Examples

  1. A box contains 8 blue and 7 yellow balls. Find the probability that one is blue and the other is yellow when two balls are drawn.
    Solution:
    Probability of getting a blue ball = P(A) = 18 .
    Probability of getting a yellow ball = P(B) = 17
    By applying the multiplication rule for independent events, 
    we get,
    P(AB) = P(A)  P(B) = 18  17 = 156.
  2. Without replacing the first card from the deck, two cards are chosen. Calculate the chances of getting a king and then a queen.
    Solution:
    We have total events = 52
    Since the first draw is not replaced, the events are dependent in this case.
    P(K) = probability of selecting a king = 4/52.
    P(Q) = probability of selecting a queen = 4/51
    Since the first card drawn has not been replaced ; therefore the total events would become 51.
    Hence, P(A ∩ B) = P(A) × P(B|A) 

= 4/52  4/51 = 1/166.
Hence, the probability of selecting a king and a queen from the deck would be 1/166.

FAQs

  1. What is the multiplication theorem for probability?
    According to the theorem, "the probability of simultaneous occurrence of two self-determining events is provided by the product of their individual probabilities."
  2. What is the difference between probability and conditional probability?
    Probability is concerned with the possibility of a specific occurrence occurring. Conditional probability considers the likelihood of two events occurring close to one another.

Key Takeaways

In this article, we have extensively discussed the Multiplication Theorem Of Probability with the help of examples. Check out the Addition Theorem Of Probability for the following topics.

Recommended Readings:

We hope that this blog has helped you enhance your knowledge regarding the Multiplication rule of probability. If you would like to learn more, check out our articles on Coding Ninjas and visit our Library for more. Do upvote our blog to help other ninjas grow. Happy Coding!

 

Previous article
Probability
Next article
Limits, Continuity and Differentiability
Live masterclass