Probability using combinatorics
Probability with counting outcomes
Find the probability of flipping exactly two heads on 3 coins. So to figure out this probability, a good place to start is just to think about all of the different possible ways that we can flip 3 coins. So we could get all tails. Tails, tails, tails. We could get tails, tails, heads. We could get tails, heads, tails. We could get tails, heads, heads. We could get heads, tails, tails. We could get heads, tails, heads. We could get heads, heads, tails. And then we could get all heads. We could get all heads over here. So there are 1, 2, 3, 4, 5, 6, 7, 8 possible outcomes. 8 possible outcomes. Now how many of the outcomes involve flipping exactly 2 heads? Let's see, that's all tails. That's 1 head, 1 head. This has 2 heads right there. That's 1 head. This is 2 heads right over there. Then this is 2 heads right over here. And then this is 3 heads, so that doesn't count. So there are 3 outcomes with exactly 2 heads. So, let me spell heads properly. 2 heads. So the probability of flipping exactly 2 heads-- And the word exactly is important, because if you didn't say exactly, then maybe 3 heads, when you flip 2 heads, so we have to say exactly 2 heads. So you don't include the situation where you get 3 heads. So the probability of flipping exactly 2 heads is equal to the 3 outcomes with 2 heads divided by the 8 possible outcomes, or 3/8. So it is equal to 3/8. And we are done.