Example: All the ways you can flip a coin

Find the probability of getting exactly two heads when flipping three coins. So let's think about the sample space. Let's think about all of the possible outcomes. So I could get all heads. So on flip one I get a head, flip two I get a head, flip three I get a head. I could get two heads and then a tail. I could get heads, tails, heads, or I could get heads, tails, tails. I could get tails, heads, heads. I could get tails, heads, tails. I could get tails, tails, heads. Or I could get tails, tails, and tails. These are all of the different ways that I could flip three coins. And you can maybe say that this is the first flip, the second flip, and the third flip. Now, so this right over here is the sample space. There's eight possible outcomes. Let me write this, the probability of exactly two heads, I'll say H's there for short. The probability of exactly two heads, well what is the size of our sample space? I have eight possible outcomes. So eight, this is possible outcomes, or the size of our sample space, possible outcomes. And how many of those possible outcomes are associated with this event? You could call this a compound event, because there's more than one outcome that's associated with this. Let's think about exactly two heads. This is three heads, so it's not exactly two heads. This is exactly two heads right over here. This is exactly two heads right over here. There's only one head. This is exactly two heads. This is only one head, only one head, no heads. So you have one, two, three of the possible outcomes are associated with this event. So you have three possible outcomes. Three outcomes associated with event. Three outcomes satisfy this event, are associated with this event. So the probability of getting exactly two heads when flipping three coins is three outcomes satisfying this event over eight possible outcomes. So it is 3/8.