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"And" vs. "Or" Probability

"And" vs. "Or" Probability

Author: Katherine Williams

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

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This tutorial compares and versus or probabilities. With an and probability, it's also called a joint probability. Here, that's the probability that both events will occur. So and both. We have to have both of them, or else it won't count.

On the other hand, there's either/or probabilities, and that's the chance that either event will occur. So it's one event or the other event. Either one will count for us.

With an or probability, it's inclusive. So as a result, it might end up being used a little bit differently from everyday life. So you want to be careful and look out for that. Something talking about what's the probability of a card being red or black, that's going to be an either/or probability. So you want to make sure we're including both of those options, that a card is either red or black. And if you think about a deck of cards, there's no other colors involved, so the probability that a card is red or black is 100%. It has to be one of those two things.

Now on the other hand, what's the probability that the card is red and black? So that time, we are talking about an and, or a joint probability. We want the chance that both events are occurring. So with red and black, well there's no cards that are red and black, because they're only one of the other. So there's a 0% chance of a card being red and black.

So simply by changing one little word, so changing from an or to an and, it totally changes our setup. Let's look at some more examples.

In this example here, we're looking at a picture with several boys in it. And it asks, what's the probability that a boy is wearing a hat and yellow? So our choices are this gentleman here, here, here, here, here, and here. So those are the boys in the picture. So we're not looking at this person or these guys in the background.

So now, what's the probability that a boy is wearing a hat and yellow? So because it's and, we want someone that's doing both things, that they have a hat on and something that's yellow on.

So here there is a hat, but no yellow, so that doesn't count. Yellow, no hat, hat, no yellow, hat, no yellow, not wearing either, hat, no yellow. So it's only this boy on the end who has both a hat and a yellow shirt. So the probability there is 1 out of the 1, 2, 3, 4, 5, 6, 7 total.

So now let's look at the second example. The second example says that what's the probability that a boy is wearing a necklace or a hat? So here with the or, we're including both / they could have on a necklace or they could have on a hat, and both of that would work for us.

So here is a hat. Here's a necklace. He's got a hat. He's got a hat. He has a hat, and he has a hat. So it's 1, 2, 3, 4, 5, 6 out of our 7 are wearing a necklace or a hat.

So now on the final example, it says, what is the probability that a boy is wearing gray or a hat? So he has on a hat, and a hat, and a hat, and a hat, and a hat.

Now, these two boys are also wearing gray. So even though they have gray and a hat, they still only get counted once. So there's 1, 2, 3, 4, 5, there are 5 boys who have on gray or a hat, so our probability is 5 out of 7.

This is [INAUDIBLE] on and versus or probabilities. It's just an introduction. Other tutorials will cover this topic more in depth.

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.