Hi there, this is Ryan Malloy here at The Worldwide Center of Mathematics. In this video, we are going to discuss how to do probability tree diagrams. So suppose that we have a deck of cards, standard 52 cards, no jokers and we've already removed the 3 of clubs and the 6 of diamonds, and we're asked some questions such as what is the probability of drawing a red faced card, so either jack, queen or king that's either diamonds or hearts. Alright, let's make a probability tree. So to start we'll first classify our cards just based on their suit. So, we've got clubs, spades, hearts and diamonds. Okay, so, let's go ahead and write down the probability of drawing one from these categories. So clubs, ordinarily it would be 13/52 since there are 13 clubs and 52 cards total. But recall that we've removed these two so that there are only 50 remaining and one of the two that we removed was a club. So instead of 13/52 what we get is 12/50. Okay, spades, again the denominator is going to be 50 but this time none of the spades have been removed so that one will remain as 30, cool? Hearts, same thing as spades, denominator is 50 and our numerator is still 13 since none have been removed. Finally we have diamonds, which is similar to the clubs, the denominator is 50, the total number of cards available and there are 12 diamonds remaining since we removed the 6. Important thing to note when you're creating a probability tree is that the probability of each of these top level events, added together should equal 1. 12 plus 13 plus 13 plus12 should equal 50 which it does. Okay, so now we'll focus down in on the things that we're actually interested in since what we want is to draw a red face card. So we'll go from here. So we have two different categories. We have face, so it's called F and non face which I'll just call N. So assuming we already have a heart and we're drawing from that to try to find the face card, there are 13 total of that suit, 3 of which are face cards. Then over here, simply 13 - 3 which is 10. We'll repeat the exercise over here. Assuming we already have a diamond, what is the probability of getting a face card? Again there are three face cards total in this suit, face, non-face, but this time we're dividing it by 12, since there are 12 diamonds in our deck. And we'll do the same over here. 12 - 3 is 9. And you can simplify these if you want to but you'll see in a second why it may actually be beneficial to not simplify these fractions. Okay, so this is essentially all the information we need since we're not interested in the probability of black face cards, we just want reds. So let's try to answer the question. What is the probability of drawing a red face card from our deck? So this is an event we want and this is an event we want. We'll calculate the possibility of each of these two events happening and then add them together. So the possibility of a heart face card is going to be 13/50, the probability of drawing a heard x 3/13, the probability of drawing a face once we already have a heart. And we'll see these 13s cancel and what we get is 3/50, great. Now, diamond face, so 12/50 is the probability that we will draw a diamond at all and assuming that we do draw a diamond, the probability of it being a face card is 3/12 and again, the 12s cancel which is why we didn't simplify earlier and we get another 3/50. So we can simply add these together to answer our question. The probability of getting a face card is just 3/50 plus 3/50 which is 6/50, which is not too difficult. We could have simply counted the number of cards in the deck, that would have fit our criteria but the advantage of using a probability tree diagram is that we can very quickly find information about other problems like if we were to ask, okay, what is the probability of drawing a diamond that's not a face card or we could rearrange it and say what's the probability of finding a black card that has an odd number on it? It's very easy to take the information we already have and just shuffle it around to meet he needs of the new question. My name is Ryan Malloy and we've just discussed how to do probability tree diagrams.