Map of life expectancy at birth from Global Education Project.

## Tuesday, April 20, 2010

### Okay, let's gamble

I was considering another diversion into philosophyland, with a post yammering about the limitations of data and reification and all that jive, but I just know y'all are out there salivating to get to the damn probability and statistics already. So okay.

As I have already said, the very idea of probability is somewhat philosophically intractable but we just have to live with that. In public health, we live in a probabilistic world as a function of the limitations of our knowledge. So there you are.

Probabilities are closely related to the idea of rates. For example, the probability of death (in a given population) during a given year is the same as the death rate (i.e., incidence of death). Of course we didn't know the death rate until after the year was over; but we often use the idea of probability in connection with a prediction about the future. That means we have to estimate the probability of dying based on prior experience, if we want to look forward. We'll be talking a lot about estimates and errors as we go on.

Probabilities can be expressed as fractions where the denominator is the total population, and the numerator the number of events. That's just a way of repeating myself -- they are similar to incidence rates. However, because incidence rates can include events that happen two or more times to the same person, the probability of the event befalling a single person can be somewhat less than the incidence rate for the event. With death, obviously, this doesn’t matter, but it could with say, ear infections. We'll see how it gets tricky sometimes, in this and other ways.

Probabilities can range from zero (no way, never happen, not a chance) to 1 (happens every time, mortal lock, death and taxes). For you gamblers out there, keep in mind -- and this is important -- that probabilities expressed this way are different from odds. A probability of .75 -- which you can also call 75% -- represents odds of 3 to 1. That's easy to figure out. Let's say there's a 25% chance of winning on a scratch ticket. That means there's a 75% chance of losing, because the percentages have to add up to 100. Something has to happen! 75/25=3. You're three times as likely to lose as you are to win, that's 3 to 1. A probability of .5, obviously, represents 1 to 1 odds. And so on. We won't go back to odds until much later, because for most of what we're going to do, it's much easier to work with probabilities.

You can do arithmetic with probabilities to figure out more complicated situations. Suppose you have a situation in which there are 3 possible outcomes, and they are mutually exclusive. That means only one of them can happen. For example, you apply to college. You might get admitted with a scholarship, you might get admitted without a scholarship, or you might get rejected. If you want to know your chance of getting admitted, with or without a scholarship, just add together the probabilities of each outcome. Again, here is a simple formula:

P(X) + P(Y)=P(X or Y)

If X means getting a scholarship, and Y means getting in without a scholarship, there you have it. As you can also figure out, if Z means not getting in at all, P(Z)=1 - P(X or Y), because one of those three outcomes must happen and the sum of all three probabilities must be 1. But remember, this only works if the events are mutually exclusive!

What if they aren't? What if we're worried about two things happening, such as you get into Harvard and you get into Yale? (No endorsement implied.) In this situation, you have to assume that the events are independent, which means the admissions offices aren't talking to each other and saying we'll take Joe and you can have Sally, which would totally mess it up. In this case, you multiply the two probabilities together:

P(X) * P(Y)= P(X andY)

We like to use "*" to mean multiplication, instead of "X" as you might have learned in school, because, obviously, we also use "X" a lot to mean "whatever," so we try to minimize confusion. We don't always succeed, as you will see. Unfortunately there are symbols that get used for two or more purposes, but I'll try to keep it to a minimum.

So, if your chance of getting into Harvard is 20%, and your chance of getting into Yale is 50% because your father went there, your chance of getting into both is

.2 * .5 = .1

That's 10%. If you want to make sure to get into Yale, ask your father to give them a million bucks.

Next time, I'll get to more complicated situations. If anything is unclear, or you have suggestions for making it more clear, please let me know.