At the math office today, two math teachers were discussing probability. Two things were surprising about it. One, it was a Saturday afternoon part of our winter break, so no teachers should have been on campus. (We’re a dedicated lot, us math teachers.) Two, the topic they were discussing was so simple, and yet, it reveals the real mind-bending character that probability has on us and our students.
Question 1: What is the probability that you draw a heart or a queen from a deck of cards?
Question 2: What is the probability that you roll a die and get a number less than 4 or an odd number?
Both questions are simple enough. The first one is 16/52 (because there are 16 cards which are hearts or queens in a deck). The second one is 4/6 (because you can roll a 1, 2, 3, or 5).
The problems are seemingly the same. Let’s now look at this problem from the perspective of a venn diagram.
(We have to subtract that last term, because we added that overlapping section twice when we took .)
Let’s apply that to our two questions:
Question 1: We have a probability of: . Clearly and . What is ? Intuitively — or using our venn diagram — we know it is 1/52. And the answer works out correctly to 16/52.
Question 2: We have a probability of: . Clearly and . What is ? Intuitively — or using our venn diagrams — we know it is 2/6. And the answer works out correctly to 4/6.
However, let’s say we wanted to calculate and (the overlapping regions) mathematically? It turns out that there is something fundamental that makes these two problems different. In question 1, the two events (drawing a heart / drawing a queen) are independent. In question 2, the two events (rolling a number less than 4 / rolling an odd number) are dependent. For the first question, you can say that while in the second problem you cannot do that.
Recall that the definition of independence of two events and is if .
Checking the first question for independence, we see that the probability of drawing a heart given that you already have a queen is 1/4, and that is the same as the probability of drawing a heart (1/4). (Similarly, the probability of drawing a queen given that you already have a heart is 1/13, and that is the same as the probability of drawing a queen (1/13).) So the two events are independent.
Checking the second question for independence, we see that the probability of rolling and odd number given that you have rolled a number less than 4 is 2/3, while the probability of rolling an odd number is 1/2. (Similarly, the probability of rolling a number less than 4 given that you’ve rolled an odd number is 2/3, while the probability of rolling a number less than 4 is 1/2.) So the two events are dependent.
The teacher who brought up this problem was grading exams, and one student had calculated . And seeing the two problems were almost identical, calculated — which, as we know, isn’t right for dependent events.
What we were discussing is how we could explain to the student that the two situations are different, even though on the surface the questions seem like they are of the same form. In other words, is there a conceptual — non mathematical — way to explain that the first question involves independent events while the second question involves dependent events? It certainly isn’t intuitive, at least not to me.