Common Core Standards: Math
7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
Two wrongs don't make a right, but in some cases, adding two rights can make a wrong, especially in the world of probability. Students should already know the multiplication rule for probability using the idea of conditional probabilities. Now, they need an addition rule.
If we have two events A and B with probabilities P(A) = 0.5 and P(B) = 0.75, then P(A) + P(B) = 1.25 should be a probability, too. But because probabilities must be in the interval [0, 1], a probability of 1.25 has no meaning.
How can that be? If we look at a Venn diagram of events A and B, P(A) is the number of outcomes in circle A divided by the total number of outcomes. P(B) is the total number of outcomes in circle B divided by the total number of outcomes. The problem is that P(A) and P(B) share some outcomes in common. When the two probabilities are added together, we've double-counted those probabilities.
See? Two rights make a wrong. (But three rights make a left!)
We've just accounted for those probabilities twice, so we can apply a simple fix to the formula. Just subtract out the number of outcomes shared between A and B.
Students should be able to use Venn diagrams and mathematical logic to understand and use the addition formula. We will use Venn diagrams to illustrate the idea and then create the formula.
The outcomes in A plus the outcomes in B minus the outcomes in A and B is equal to all of the outcomes in A or B. In terms of probabilities, we can write this as:
P(A) + P(B) – P(A and B) = P(A or B)
You have a box of fruits and vegetables. The fruits (F) are apples (A) and oranges (O), while the vegetables (V) are carrots (C) and broccoli (B). You already know P(A), P(O), P(V) and P(C), which represent the probabilities of drawing that particular fruit or vegetable from the box. What is the probability of selecting a fruit?
Applying the addition equation, we have
P(A) + P(O) – P(A and O) = P(A or O) = P(F)
Of course, P(A and O) = 0, since a fruit is either an apple or an orange. There are no hybrid Franken-fruit in this box. If we made a Venn diagram in terms of the fruit, we'd see that A and O share no space. The circles don't intersect.
For students, a working knowledge of the addition rule includes recognizing that adding probabilities may result in double-counting outcomes if the two groups that share outcomes. Once students recognize this, they should be able to apply the addition formula, and hopefully even derive it for themselves.