Explanation of conditional probability: If X happens, then...

- Suppose I have the names of six people, and I put one person's name on each of six cards. I'm gonna pick two cards at random and award each person whose names are on those two cards $100. Let's say Jose and Sally are two of the six people on my list, what are the odds I will pick both of their names? Well, here are the 15 possible outcomes, only one of which contains both Jose and Sally. So, the probability that both will win together is 1/15 or 6.67%.

Let's say I change the way we play the game. Let's say that instead of picking both cards at the same time, I will pick one name and then pick the second name a few minutes later. What are the odds of both Sally and Jose winning if the first card I pick has the name Audrey? For this, we use the concept of conditional probability. This helps us answer questions like this one. In this case, now that Audrey is one of the winners, it is impossible for both Sally and Jose to win.

The probability that both Sally and Jose are the two winners has dropped from 6.67% before we picked Audrey's name to zero after Audrey's name was picked. But, what about if Sally's name was on the first card? What are the odds that both Sally and Jose win now? Remember these were all of the possible outcomes. But now that Sally is out of the pool of names, here are the five possible outcomes that remain.

As we can see, initially the odds that Sally and Jose both win were 6.67%. Once Sally's name is chosen, the probability went up to 20% since Sally and Jose were one of five possible outcomes. Often it helps to draw probability trees to visualize what's happening. Here's a probability tree for three coin flips. As you can see, the odds of getting tails three times in a row is initially 12.5%.

We get that by multiplying the probabilities along this branch. Once we flip, the first tails, though, the odds of getting tails three times in a row increases to 25%, only two parts of that branch remain. After the second tails is flipped, then the odds jump to 50%. Let's look at one more problem that does not include cards or coins. Here's a set of health-related data: 1,000 people, how long they lived, and whether or not they exercised at least three days per week.

Consider these two events: event A, people that lived more than 85 years, event B, people that exercised at least 30 minutes three or more days per week. How would we find the probability that someone lived more than 85 years given that they exercised at least three days per week? Let's build the tree for this scenario. The given event is that this person exercised three days per week.

So this is like the first coin flip. 240 out of 1,000 people exercised three days per week. 760 of 1,000 people did not exercise at least three days per week. Now comes the second event, how long did they live? 40 of the 240 exercisers lived less than 75 years, 16.6%. 70 of the 240 exercisers lived 75 to 85 years, 29.2%.

And 130 of the 240 exercisers lived more than 85 years, 54.2%. Here's what the other branch on that tree would look like. So, from all the data we can see that only 13%, 130 of the 1,000 people lived more than 85 years and exercised at least three days per week. But once we are given the fact that the person in question worked out at least three days per week, the probability that this person lived more than 85 years is 54.2%.

According to this data, it looks like exercising three days per week might have its advantages. I guess we can say that working out is an enormous factor in living past 85. Not so fast. Don't go overboard. People who are committed to working out three days a week likely have other good habits. So, the exercise alone may not be the only contributing factor. Understanding how to calculate conditional probabilities is very important in the world of statistics.

But understanding what those numbers might mean and knowing which questions to ask, that might even be more important.