General Addition Rule / General Multiplication Rule

The general addition rule finds the probability of an event meeting either of two separate criteria. So let's say on my low carb site I polled 1,000 random readers who were in a relationship. I asked each reader where their BMI scores put them - underweight, healthy, overweight, or obese. Their answer would be one of those four categories. I also asked them what category their partner fell into. So that is a separate set of category responses.

Now let's say I wanted to figure out the probability of a person or their partner being obese. I would want to use the general addition rule to do that. The probability of a couple having the respondent being obese or their partner being obese would be equal to the probability of the respondent being obese, plus the probability of the partner being obese, then minus the probability of the joint event of "both people are obese" because those joint events would have shown up in both initial numbers (i.e. double counted). This would then tell me the probability of a given reader in the sample set either being obese or having an obese partner.

The general multiplication rule is a different situation. There are two flavors. Let's say you are looking at independent events. This is when you are trying to figure out that two separate events are both true. So in the above example, let's say I'm curious to find out how many couples in my sample set are both obese. I want to find responses where the respondent is obese AND their partner is obese too. The probability of both people being obese is equal to the probability of the respondent being obese times the probability of the respondent's partner being obese.

You can also use general multiplication for other purposes. Let's say I wanted to look at the probability that two separate respondents are both obese. For this you calculate the probability of the first respondent being obese. Now they are counted and are 'out of the pool' so you calculate the second person's probability without re-counting that first person. So if there was a probability of 500/1000 that the first person was obese, the probability of the second person being obese - since we've already processed and counted that first person - is now 499/999. You multiply those numbers together to get the overall chance that both respondents are obese.

To Describe Further:

Let's say we are looking at 100 people and we are curious about the pets they own. We come up with this grouping:

general addition rule

So the whole box is the 100 people (I'm a very visual person). Now some of them have cats. Some of them have dogs. Some of them have cats AND dogs. So the red circle is the cat owners. The blue circle is the dog owners. The overlap is the people who have cats AND dogs. The white area are the people who don't have cats or dogs.

So far so good?

Now the General Addition Rule is about figuring out who has A *or* B. So you are trying to figure out who is in the red circle OR who is in the blue circle. Your total number is going to be everyone who is red - as well as everyone who is blue. You're trying to count everyone who is "colored" on this page. Normally you would count everyone in the whole red circle. Then you would count everyone in the whole blue circle. But look at the page! The people who were in the darker middle area would have been counted two times, right? They would have been counted once for being in the red circle, and then counted a second time for being in the blue circle. So you have to account for that. In the end you want to count every person who is in that pair-of-circles thing in the middle of the white box. You count every person who meets either criteria.

In comparison, the General Multiplication Rule is for only counting the people who are A **and** B. You don't care about people who just have cats. You don't care about people who just have dogs! You don't care about people with no pets at all. The only people you care about are the people in the center dark slice - the people who have cats AND dogs.

Now part of this is that you can use the General Multiplication Rule for complex kinds of questions. For example, if we choose a person who has a cat what's the chance that the next person we choose also has a cat? But you can also use it for asking straight "and" questions, for example, how many of those 100 people in Sutton have both a cat and a dog.

I think the confusion might happen because of the word "and". You could say, with the Addition rule, that you are looking at people who are red AND people who are blue. You are adding up the red and blue circles. But in the world of unions and intersections that would be confusing, because in that world the word "and" means they have to be both things. They have to be red AND blue. They have to be that purple area in the middle. What you are really saying they can be red OR they can be blue. Either case is OK. That way it's clear that you're not aiming for the group in the middle who are both things at once.

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