In this video, learn how to find the probability of two events that might or might not intersect and when to add and subtract probabilities.
- Sometimes there are multiple outcomes that would lead us to the same conclusion. Let's play a game. We'll flip two coins. If even one coin comes up heads, you win. What are your odds of winning? Well, here are the four possible outcomes: heads on both the first and second coin, tails on the first and second coin, and so on. Let's call each coin flip an event. So if event one is heads, we win. Or if event two is heads, we win. And if both events are heads, well, we're hot. We're on a roll. Our luck is good. Vegas, here we come. Notice, we have an overlap. When we get heads on both flips, there's a winner on event one and a winner on event two. We want to be sure not to double-count that outcome. So 50% of our four outcomes provide a winner on flip number one. 50% of our four outcomes provide a winner on flip number two. And we then subtract the overlap, where we had winners on both flips. This was on 25% of the outcomes. So the probability of getting heads on at least one of two coins is 75%. This was an example of what we call the addition rule in action. Add up the probabilities of each desired outcome, and subtract away the overlap. This was an easy problem, so let's up the ante. Let's say we have a pair of six-sided dice. To win, roll a six with either die number one or die number two. Each die has six sides, so each number on die number one can be matched with six different numbers on die number two. 6 sides time 6 sides equals 36. So there are 36 possible outcomes. Here are all 36 outcomes. In six outcomes, we roll six on die number one. In another six outcomes, we roll six on die number two. But if we roll six on both dice, it overlaps both of these groups. Let's use our addition rule. There are six outcomes where event number one results in rolling a six and six outcomes for event number two, but one outcome overlaps. 6 plus 6 minus 1 gives us 11 total outcomes. 11 outcomes of the 36 possible outcomes is 30.56%. Let's try one more. What's the probability that our two dice will add up to either 7 or 11? Here are the six outcomes where the dice add up to seven. Here are the two outcomes where the dice add up to 11. But notice there is no overlap, so here there is no need to subtract anything. Six plus two gives us eight total outcomes. 8 outcomes out of a possible 36 outcomes is 22%. These were all relatively simple scenarios. Therefore, illustrating all the possible scenarios was not difficult. But in more complex scenarios, you may need to call upon the permutation and combination skills you developed in chapter one. But don't forget today's lesson. In any single scenario, you can add up the probabilities just as long as you remember to subtract out the overlapping outcomes.
Eddie explains that probability is used to make decisions about future outcomes and to understand past outcomes. He covers permutations, combinations, and percentiles, and goes into how to describe and calculate them. Eddie introduces multiple event probabilities and discusses when to add and subtract probabilities. He describes probability trees, Bayes’ Theorem, binomials, and so much more. You can learn to understand your data, prove theories, and save valuable resources—all by understanding the numbers.