The Law of Total Probability | Probability Theory, Total Probability Rule

What is the law of total probability? Also sometimes called the total probability rule, we go over this tremendously useful law in today’s full video math lesson! Imagine we have a sample space that can be partitioned into three events B1, B2, and B3. And say we have another event in this sample space called A. Then, whenever A occurs, it must be true that B1 also occurs, or B2 also occurs, or B3 also occurs, by definition of a partition. It also follows from the definition of partition that B1, B2, and B3 are disjoint, no more than one of them can occur at the same time.

Therefore, we can find the probability of A by adding the probability that A and B1 occur, the probability that A and B2 occur, and the probability that A and B3 occur.
That is, P( A ) = P( A intersect B1 ) + P( A intersect B2 ) + P( A intersect B3 ). This is the law of total probability for a partition into three sets, but it is also true for any partition into a finite number of sets, or a countably infinite number of sets!

So if we know the probabilities of those intersections, we’re good to go, but often times we do not. Instead we use the formula for the probability of an intersection that uses conditional probabilities! Recall the definition of conditional probability, P( A | B ) = P( A intersect B ) / P( B ), which gives us the equation P( A intersect B ) = P( A | B )*P( B ). So if we know the probability of A given certain conditions, we can use this equation and apply the law of total probability.

The law we previously wrote as: P( A ) = P( A intersect B1 ) + P( A intersect B2 ) + P( A intersect B3 ) then becomes:
P( A ) = P( A | B1 )*P( B1 ) + P( A | B2 )*P( B2 ) + P( A | B3 )*P( B3 ) .


SOLUTION TO PRACTICE EXERCISE:

We are given that 40% of students are male and 60% are female. Sixty percent of males are taller than 6 feet, meaning 40% of the males are shorter than 6 feet. Ten percent of the females are taller than 6 feet, which implies 90% of the females are shorter than 6 feet. We want to find the percent of the class that is shorter than 6 feet.

The sample space is partitioned into males and females. Let E be the event that a student is shorter than 6 feet, M be the event that a student is a male, and F be the event that a student is female. Then, from the law of total probability, we have that P( E ) = P( M intersect E ) + P( F intersect E ). In this case we don’t know the probabilities of these intersections but we do know some conditional probabilities. Rewriting the equation using conditional probabilities, we have that P( E ) = P( E | M )*P( M ) + P( E | F )*P( F ). Thus, P( E ) = 40%*40% + 90%*60% = 16%+ 54% = 70%.


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If you are preparing for Probability Theory or in the midst of learning Probability Theory, you might be interested in the textbook I used when I learned Probability Theory. It is "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/2Z8MrB7
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I hope you find this video helpful, and be sure to ask any questions down in the comments!

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The outro music is by a favorite musician of mine named Vallow, who, upon my request, kindly gave me permission to use his music in my outros. I usually put my own music in the outros, but I love Vallow's music, and wanted to share it with those of you watching. Please check out all of his wonderful work.

Vallow Bandcamp: https://vallow.bandcamp.com/
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