Probability OR Explained

In probability, two events can be linked with conjunctions like AND or OR. In this post, we’ll explore the probability OR, and explain how to calculate it.

Maybe you’re thinking, I know what “or” means! But when we’re talking about probability, that little word has a very specific meaning, and it’s not always the same as the regular English meaning.

Probability OR: What it Means

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In English, we often use “or” when we mean “one or the other but not both.” For example, the server in a restaurant might ask you if you want “soup or salad” with your meal. If you say “both,” there will be an extra charge!

In the world of probability, though, OR means “one or the other… or maybe both.” It’s not an exclusive or, the way it often is in regular spoken English, where choosing one means you don’t get the other.

Instead, you could have both of the events and it still counts as OR. Think of it as “at least one of these options.”

An Example of OR

For example, maybe you’ve been watching two of your co-workers, John and Rhonda, and you notice that they both seem to wear blue a lot. You’re interested in the likelihood of at least one of them wearing blue on any given day.

Let’s name two events: J will be the event “John wears blue” and R will be “Rhonda wears blue.”

Then the probability that at least one of them wears blue is written P(J OR R). The only way “J OR R” wouldn’t be true is if both of them were not wearing blue.

Probability OR: Calculations

The formula to calculate the “or” probability of two events A and B is this: P(A OR B) = P(A) + P(B) – P(A AND B).

To see why this formula makes sense, think about John and Rhonda wearing blue to work. Suppose John wears blue 3 out of 5 days each week, so his probability of wearing blue is 60%. Let’s say Rhonda wears blue 4 out of 5 days a week, so her probability is 80%.

If we just added those together to combine the probabilities, we would have a probability of 140%. Obviously, that’s not possible! The problem is that there are some days that they both wear blue, and we’re counting those days twice.

So, the formula includes the last, subtracted term to make up for that. We need to subtract off the probability that the two events overlap. That overlap is being counted in the P(A) and in the P(B), so we need to remove it once to have an accurate probability.

OR Example, Continued

In our John and Rhonda example, we have P(J) = 0.6 and P(R) = 0.8. Suppose we knew that they both wear blue on about half the days, so that P(J AND R) = 0.5. Then we could calculate P(J OR R):

P(J OR R) = P(J) + P(R) – P(J AND R) = 0.8 + 0.6 – 0.5 = 0.9.

That tells us that the probability that at least one of these two co-workers is wearing blue is a staggering 90%! Always double check that your answer to a probability question is between 0 and 1.

Mutually Exclusive Events

One special case to mention is mutually exclusive events. When A and B are mutually exclusive, they can’t happen at the same time, so P(A AND B) = 0.

Then the formula for the OR probability becomes P(A OR B) = P(A) + P(B). But remember, this is only for mutually exclusive events!

Be sure to check out our probability and statistics video lessons for more topics from the world of probability and statistics!

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