Algebra II—Semester B

Sample Lesson - Introduction

Lesson 11.04: Theoretical Probability: AND versus OR

Really? You don't have the popped kind? What are we, squirrels?

(Source)

If we're going to totally rely on the power of our calculations when we do math, we had better be confident in our ability to model every conceivable situation. For instance, say we're at the movies with two friends and they ask us to purchase the snacks. Our first friend is pretty easygoing. She'd be fine with Jujubes or Raisinets or Milk Duds, but our other friend is kind of greedy. He wants Jujubes and Raisinets and Milk Duds. It seems like the probabilities that either friend will be happy with what we bring back are pretty different, but why?

What's causing this whole problem lies with two little words: "and" and "or." They may be short, tiny words, but their impact is huge, especially when it comes to probabilities. When we say "and," we're limiting our options. We're making the desirable specific event more specific. When we say "or," we're opening up the specific event to more possibilities. There's a greater number of outcomes that would make us happy.

This lesson's reading will detail how our formulas must change to accommodate these different scenarios. Keep in mind that when we use "and," our probability will go down, and when we use "or," it will go up. You're more likely to be happy if you're open-minded about the things that can make you happy. There's a lesson in there somewhere. Lower your standards?

Sample Lesson - Reading

Reading 11.11.04: Either/Or (Or And)

We're familiar with the word "and" from our experiences as speakers of the English language. It's often used to connect two things that are both true: if some guy says, "Today I'm going biking and swimming," then he'd better end up on a bicycle and in a large body of water before midnight, or he'll be a bit of a liar.

Talking About "And"

In probability, we use some fancy symbols to express pretty much the same idea. For two events A and B, we write the probability of A and B as P(A ∩ B). That hump is called "the intersection" of A and B, but we'll sometimes just put it in English: P(A and B). It's all the same.

Whatever we call it, this is the probability that both A and B occur. So if we're trying to calculate P(A ∩ B), we count all of the outcomes that satisfy both A and B, and divide this number by the total number of outcomes. Sound simple? It is.

Say we have a deck of cards, and we're curious what the probability of randomly drawing a red king is. We've got two events, A and B; the first, A, stands for "the card is a king," while B is "the card is red." We're trying to figure out P(A ∩ B). So, we count how many outcomes satisfy both A and B.

In a standard 52-card deck, there are only four kings, and only two of those kings are red: the king of hearts and the king of diamonds. With 52 possible cards to pick from, .

This process gets even easier when A and B are mutually exclusive; that is, if A and B can't both occur at the same time. If that's the case, then there's no way for them to both occur, so no matter what A and B are, P(A ∩ B) = 0.

We can see this if we look back to our deck of cards. Being a king and being red aren't mutually exclusive, but being a king and being a queen are. If we try to calculate P(A ∩ B), where A means "the card we draw is a king" and B is "the card drawn is a queen," then we can be confident that P(A ∩ B) = 0. (We almost want to say that you can bet on it, but Shmoop doesn't gamble, even when the probabilities are indisputable.)

How About "Or"

"Or" is a different ballgame, but one that also borrows some of its rules from plain old English. It's often used to connect two things when only one of them needs to be true. If that same guy from earlier says, "Today I'm going biking or swimming," then all he needs to do to be speaking the truth is spend some time biking, or some time swimming. He could even do both, and we wouldn't hold it against him.

The same general idea applies when we're talking in terms of probabilities. For two events, A and B, the probability of A or B is called the union of A or B, P(A ∪ B). It's the probability that A will occur, or B will occur, or both will occur.

"Or" is kind of like the bus of probability, because it gives everything a ride. We just add up all of our events and throw them up top in our probability fraction, over the total number of outcomes. Let's grab our handy deck of 52 cards and take "or" out for a spin.

What's the probability that we grab a queen or a red card? There are four queens. There are 26 red cards (half of the deck). So we just add those up and head to Fractionland, right?

Not so fast.

We accidentally counted the red queens twice: once when we counted queens, and once when we counted the red cards. We need to subtract the overlap (the two red queens) from our total before we fraction it up. We end up with 28 cards that are red or queens. Our probability then is .

In terms of a formula (formulas are always good), P(A ∪ B) = P(A) + P(B) – P(A ∩ B). This is pretty much just a mathematical translation of the ideas we've been teasing out: add the probability of one event to the probability of the other, make sure to subtract any doubled outcomes, and you're done. It's a bit more elegant as a formula, though.

Recap

The intersection of two events, A and B, is the probability of both of them occurring together. In symbols, that's P(A ∩ B).

To calculate P(A ∩ B), we count all of the outcomes that satisfy both A and B, and divide this number by the total number of outcomes.

When A and B are mutually exclusive, P(A and B) = 0.

The union of A and B refers to the probability that A occurs, B occurs, or they both occur. You'll recognize it on the street by its notation, P(A ∪ B). If you want a fancy formula for calculating the union (and you know you do), use this one: P(A ∪ B) = P(A) + P(B) – P(A ∩ B).

Sample Lesson - Activity