The Addition Rule of Probability | Probability Theory, Sum Rule of Probability

What is the addition rule of probabiltiy? Also sometimes called the sum rule of probability, this rule tells us how to calculate the probability of the union of two events. In today’s math video lesson, we’ll explain the addition rule of probabiltiy for two events that are mutually exclsuive and for the general case when we don’t know if the events are mutually exclusive! We’ll also go over two examples of addition rule problems. It’s called the addition rule because, of course, it involved adding probabilties!

If A and B are two mutually exclusive events, then P(A U B) = P(A) + P(B). This is the addition rule for mutually exclusive events!

But what if we don’t know if two events are mutually exclusive? Or what if we know they are certainly not mutually exclusive? Then we can use this formula instead: P(A U B) = P(A) + P(B) - P(A intersect B). This comes from the fact that counting P(A) also includes P(A intersect B) by definition. However, when we add P(B), we are counting P(A intersect B) a second time, so we need to subtract P(A intersect B) in order to correct that double counting of the intersection! When two events are mutually exclusive, the probability of their intersection is 0, so we’d just be left with the original formula for the probability of the union of two mutually exclusive events.

Remember that mutually exclusive events are events that cannot occur simultaneously!

SOLUTION TO PRACTICE PROBLEM:

We are asked to find the probability of a randomly drawn card from a standard 52-card deck being red or an ace. We are considering two events.

A: The card is red
B: The card is an ace

A and B are not mutually exclusive because a card can be red and an ace (ace of hearts or ace of diamonds). We know, by the addition rule, P(A U B) = P(A) + P(B) - P(A intersect B). What is P(A)? There are 26 red cards and 52 cards total, so P(A) = 26/52. What is P(B)? There are four aces so P(B) = 4/52. What is P(A intersect B)? There are two red aces, so P(A intersect B) = 2/52. Thus, P(A U B) = 26/52 + 4/52 - 2/52 = 28/52. 

If you are preparing for Probability Theory or in the midst of learning Probability Theory, you might be interested in the textbook I used in my Probability Theory course, called "A First Course in Probability Theory" by Sheldon Ross. Check out the book and see if it suits your needs! You can purchase the textbook using the affiliate link below which costs you nothing extra and helps support Wrath of Math!

PURCHASE THE BOOK: https://amzn.to/31mXEjr


I hope you find this video helpful, and be sure to ask any questions down in the comments!

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