 independent events with rolling a die and flipping a coin. So today we're going to talk about some probability and we're going to discuss independent events. In probability, when you are able to work with independent events, it's, there's some real advantages to it that we'll get into. But here is a question. Suppose we flip a coin and we roll a die, standard six-sided die, and we want to know the probability of getting a head on the coin and a six on the die. So to go about this, when your probability is success over total number of outcomes. So let's look at the number of outcomes. So let's start by rolling our die. We can roll the die and get a one, a two, a three, a four, a five, or a six. And then after we roll that die, we can flip heads or tails, heads or tails, and so on. My diagram here might not be the prettiest, so I decided to make a nicer one and go ahead and embed it into my slideshow here. So here you see you can roll a dice and get a one, two, three, four, five, or six. And then you can flip that coin and get heads or tails, heads or tails, heads or tails, heads, tails, heads, tails, heads, tails. So if you're looking at the total number of outcomes, you can flip ahead and roll a one, flip tails, and a one, so on and so forth. So I have all 12 outcomes listed here. And I decided to go ahead and list them out here in what we call a sample space. A sample space is simply a listing of all possible outcomes. So you can count up here there are 12 outcomes. Drawing a diagram is fine, but sometimes drawing the diagram can become cumbersome. It can become labor intensive when we start looking at perhaps rolling a dice four times, or rolling four dice, or perhaps flipping a coin 20 times. Drawing a diagram can become very cumbersome. So first let's talk about, is this an independent event? As in, when I flip a coin, does it care what happened with my die? Does the coin care that I just rolled a two, or does I just roll a three? When the coin is in the air, is it thinking to itself, oh, I just saw a two pop up. I need to land on heads, or I need to land on tails. The coin does not care how the dice performed. The dice does not care how the coin performed. These are independent events. Because these are independent events, we can do something very, very helpful here. So you see, when you roll a dice, there are six outcomes. When you flip a coin, there are two outcomes. And so we can see another way to arrive at 12 would have been six times two. The number of outcomes for the first event times the number of outcomes for the second event. We're interested here, let's see if I can find the question. We're interested here specifically in flipping a head and rolling a six. So heads and six. This would be one possible outcome out of 12 possible outcomes. The probability of rolling a six and then flipping a head would be one out of 12. And while we're here, let's think about this yet another way. These are independent events. The coin does not care how the dice performed. Versa vice so the dice doesn't care how the coin performed. The probability of rolling a six would be one out of six. The probability of flipping a head would be one out of two. So another way we could have approached this would have been to say the probability of the first event times the probability of the second event gives us the probability of the two events together. And this is an example of independence.