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"And" vs. "Or" Probability

"And" vs. "Or" Probability

Author: Ryan Backman

Calculate the probability of an "and" or "or" event.

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Hi. This tutorial covers "and" versus "or" probability. OK. So let's start with a chance experiment. So consider a chance experiment where a political organization is interested in the marital and children status of adults in a certain neighborhood. So marital status outcomes are married, which we'll abbreviate with an M, single, S, divorced, D, widowed, W. And children status outcomes are children, C, or no children, NC.

So let's use a diagram to make the sample space. So remember with a tree diagram, we always start kind of at the top at a node. OK. Now, the first thing that we're going to look at when we sample somebody is their marital status. So we'll have-- there's four different outcomes here. So we're going to have four branches. And your outcomes were married, single, divorced, widowed. OK.

Now based on that, there are two outcomes coming from each of these-- so children or no children. So we can see that there are eight total outcomes for this chance experiment.

If we look at this outcome here, that person would be married with no children. If we look at this, that would be divorced with children now. If we look at the last outcome here, that would be widowed with no children. OK. So we can see that there are eight possible outcomes, and they're displayed pretty easily using the tree diagram.

So now, an adult from the neighborhood in question is chosen at random and is asked about his or her marital and children status. So we're going to let A be the event that the adult is married. OK. So we want to know, well, what's the-- excuse me-- what are the possible outcomes for event A?

Well, all we're concerned with is that the adult is married. So we want to know, well, how many of the eight outcomes represent somebody that's married? And it's going to be this outcome and this outcome. So there are two outcomes. And the way we'll notate those are MC and MNC-- married with children, married no children. OK.

Let B be the event the adult is married and has children. So what are the possible outcomes here? Married and children. So now both of those two events-- or both of those two outcomes need to be that-- both married and children. Well, there's only one out of all the outcomes where it's married with children. OK. So the possible outcome there, there's only one of them-- MC.

OK. Now let C be the event that the adult is married or has children. What are the possible outcomes here? So now, married or has children. So in this case, married with children, that works. Married, no children, that works because, in that case, both are married-- both of these outcomes include somebody that's married.

And then single with children, divorced with children, widowed with children are all-- would all be outcomes where the person would be either married or have children. So if we write those out-- MC, M no C, so married not children, single children, divorced children, widowed children. OK. So there are five possible outcomes here.

You'll notice that I included married and children in this situation. Whenever we're dealing with "or"-- when we're dealing with probability is that's going to be what we call an inclusive "or." So the inclusive "or" contains both the "and" outcomes and the "or." So some people think of "or" as and/or.

All right. There's also a couple symbols here that I think are important to introduce. A and then there's this upside down U and then B. So what that means is A and B. That's like the example where we said married and children. And it's also sometimes known as the intersection of A and B. So both outcomes need to be met there.

OK. Now, if we're dealing with A or B, this U symbol here is known as the union. So that's A or B, sometimes known as the union of A and B. So you going to see either using the symbol or using either of these two symbols or and or or there.

So and then this is the recap we talked about before. In probability, "or" is an inclusive "or." The outcomes in A or B include the outcomes in A or B or both. OK. So that's important there.

And as we move forward with probability, we're going to be using "and" and "or" quite a bit in probability. So if you're dealing with a joint probability, a joint probability is the probability of two events occurring. Sometimes this is known as an "and" probability. So we won't do any calculations here. But if we wanted to know, well, what's the probability of selecting somebody that's married and has children, whatever probability we come up with, that's going to be considered a joint probability.

And then an either/or probability is the probability of either of the two events occurring or both occurring. This is generally your "or" probability. So if we wanted to know what's the probability of somebody being married or having kids, that would be called an either/or probability. So that has been the tutorial on "and" versus "or" probabilities. Thanks for watching.

Terms to Know
"Or" Probability

The probability that at least one of two events, A or B, occur.

Joint Probability/"And" Probability

The probability of two events A and B both occurring.