Multiplication Rule of Probability | Probability Theory, Intersection of Two Events

Video Description

What is the multiplication rule of probability? How do we find the probability of the intersection of two events? That’s what we’ll go over in today’s probability video lesson! The multiplication rule can also be generalized to apply to the intersection of more than two events, but we’ll talk about that in another lesson! If two events, A and B, are independent, then P(A and B) = P(A)*P(B). This is fairly intuitive when we look at some examples like the flipping of two coins for example. What is the probability that the first coin lands heads up? That would be 1/2. What is the probability that the second coin lands heads up? That is also 1/2. The events are independent. Flipping heads on the first coin does not change the probability of the second coin landing heads up. What is the probability of both coins landing heads up? That would be (1/2)*(1/2) = 1/4. However, things get more complicated when we deal with dependent events. This is because if an event A has occurred, and it is not independent from B, then the probability of B is affected by the occurrence of A. Check out the full video lesson for an explanation of this situation, and how the multiplication rule applies! The multiplication rule in this situation uses what is called a conditional probability. Check out my lesson on conditional probability: https://www.youtube.com/watch?v=FtFR80FcJjU SOLUTION TO PRACTICE PROBLEM: There are two ways that one ball from the jar can be green and one can be red. Either the first is red and the second green, or the first is green and the second red. A: First ball is red B: Second ball is green Then P(A) = 7/17 (number of red balls over total number of balls) and P(B | A) = 10/16 (number of green balls over new total number of balls). Thus, P(A and B) = (7/17)*(10/16) = 70/272. We can calculate the probability of the first ball being green and the second being red in a very similar way, and it is exactly the same probability. Try verifying this fact yourself. Thus, the probability that one ball is red and one is green is 70/272 + 70/272. = 140/272. To check our answer, we could calculate the probabilities of the rest of the possible outcomes of drawing two balls (the other possibilities are that they are both red or that they are both green) and then add those probabilities to 140/272. The total sum should be 1. Try checking your answer this way! 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! ******************************************************************** 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/ Vallow Spotify: https://open.spotify.com/artist/0fRtulS8R2Sr0nkRLJJ6eW Vallow SoundCloud: https://soundcloud.com/benwatts-3 ******************************************************************** +WRATH OF MATH+ ◆ Support Wrath of Math on Patreon: https://www.patreon.com/wrathofmathlessons Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu My Music Channel: http://www.youtube.com/seanemusic