 and welcome back. Today what we're going to do is we're going to talk about theoretical probability. So theoretical probability is the probability that an event will happen. Now these probabilities, they range from 0 to 1, so a probability of an event, it goes from 0 to 1, where 0 is that the event is not going to happen and then 1 is there's 100% chance that it's going to happen. So most of these probabilities that we're going to be finding are going to be fractions in between 0 and 1. Now the probability event is always the outcomes that we want over the total number of outcomes that could possibly happen. So we're going to look at these two couple of examples to get a better idea of what theoretical probability is. So each letter of the word probable is written on a separate card. The cards are shuffled. What is the probability that a randomly selected card is a consonant? So we need a little bit of English knowledge here. So what we're looking for is the probability of drawing a consonant out of this deck of cards that we've made. So now when you write your probabilities, when you start with your notation here, p stands for probability and an in parentheses here is simply just a word that you're going to use to help you, a word to help with whatever probability you're finding. In this case, we're looking for consonants. We're looking for consonants, so that's the word that I use here. Now in English, in a word, we have consonants and vowels. Vowels are the O, the A, and the L. The consonants are everything else. P, R, B, B, and L are my consonants. So those are the outcomes that we want. So if I randomly select a card, I want P. I want R. I want B. I want B or I want L. One of these one, two, three, four, five different cards. Those are the ones that I want. Let me write this a little different. This is just a five there. Let's do wants. So I usually abbreviate this. Want over total is what I usually use to abbreviate my probabilities. So in this case, I want those five cards for a total of one, two, three, four, five, one, two, three, four, five, six, seven, eight, eight total cards. So there is my probability. Five out of every eight cards that you draw theoretically are going to be consonants. Next one down here, two number cubes. Now when we talk about number cubes, we're talking about six-sided die. We're talking about those cubes that you see in the movies when they throw at the roulette table or something like that. That's what we mean by number cubes. They have the numbers one through six on them. So I get another number cube out here. The numbers one through six, and interestingly enough, cool fact about the number cubes is that the numbers on one side and then the number on the complete opposite side always add up to seven. So notice the one and the six, those are on completely opposite sides of the cube. Anyway, so two number cubes, numbers one through six are rolled. What is the probability that the difference between the two numbers is four? What's the probability that the difference between the two numbers is four? Now what I'm going to do is this one's going to involve just a little bit more math other than just what we used up here. So this is a little bit more difficult problem. But the notation stays basically the same. Probability of rolling two number cubes was the probability that the difference between the two numbers is four. So I'm looking for a difference, difference of four. Now notice I just abbreviated it. Again, what's in here, what's in the parentheses, is just used as a way for us to kind of understand when we're writing the problem what we're actually looking for. We're looking for the difference of four. So in this case, what numbers give me a difference of four? Well, when I roll these dice, a six and a two, that's going to give me a difference of four. So that works. Also, when I roll these die, a five and a one will also give me a difference of four. So what I'm doing here is I'm actually writing on top here the different combinations of dice that are going to give me this difference of four. I'm just writing them out. These are what I want. This is what I want here on top. Instead of writing numbers, I'm drawing pictures to get a better understanding of what's going on. Now, but here's the thing. I'm rolling two number cubes. I can roll a six first for one number cube and a two for the second number cube. Well, actually I can also roll a two for the first number cube and I can roll a six for the second number cube that would also give me a difference of four. And same vice versa with this other one. I could also roll a one first and then I could roll a five second. This would also give me a difference of four. Interesting. Now, the thing is when we're looking at these type of problems, you have to think of all the possible different combinations that we can have. All right. Now, as I'm working with this, now this is what I want. These are the four different combinations of dice that would work for me. Now, what I need to know on the bottom here is the total. Well, the first dice that I roll is going to have six different numbers that could happen. And then the second dice that I roll is going to have six different numbers that can happen. So what I'm going to do is I'm going to use my fundamental counting principle and I'm going to take six times six, six choices for the first dice times six choices for the second dice to get my total number of possible events that could happen, my total here. So this is going to be four over 36. Four possible combinations of dice that give me a difference of four over a possible 36 dice rolls that could happen for a total of reduce this number to, what is that, one over nine. So one out of every nine tosses of these two number cubes is going to give me a difference of four. So there we go. Now, the reducing, the fractions in reducing is pretty straightforward. It's the notation here and then the coming up with your what you want over your totals, which can be a little bit difficult. Okay, that's a short video on theoretical probability. Thank you for watching and we'll see you next time.