“Or” / “and”

One of the formulas typically taught in mathematics is

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

In ordinary English, the probability that either event $A$ or $B$ happens is the probability of event $A$ plus the probability of event $B$ minus the probability that the both occur.

For example, when rolling two fair six-sided dice, the probability that at least one three appears is

$P(A \cup B) = \displaystyle \frac{1}{6} + \frac{1}{6} - \frac{1}{36} = \displaystyle \frac{11}{36}$.

It’s necessary to subtract something off at the end because it’s possible for the first die to be a four and simultaneously the second die to be a four.

This can be a conceptual barrier for students if it’s not directly addressed. In mathematics, the word “or” means “one or the other… or maybe both.” In the previous example, event $A$ was “first die is a four” and event $B$ was “second die is a four,” and it’s possible that both events could occur simultaneously.

Of course, this is different than the way we typically use “or” is spoken English. For example, in the final episode of each season of “The Bachelor,” the guy has to choose one woman or the other… and there’s no possibility of him choosing both! When a student says, “Next semester, my morning class will be history or physics,” we don’t think that there’s a possibility that the student will choose both classes… the student will choose one or the other, but not both.

In terms of mathematical logic, the word “or” in ordinary speech really is an “exclusive or.”

As I said, this isn’t a big deal for students to see, but in my opinion it’s best to directly address this subtlety rather than have students confused about which meaning of the word “or” they should be using when doing their homework.

P.S. The good news is that the word “and” means the same thing in the language of probability/logic as its meaning in ordinary speech.

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