Three Types of Probability
This article is not so much about particular problems or problem solving tactics as it is about labels. If you think about it, labels are a big key to the way we organize ideas. When we alreadyhave the central concepts to problems organized, we are better able to solve them and our solutions are often more eﬃcient. In short, labels help us organize – and organization simpliﬁes problem solving! This article seeks to demonstrate the power of intelligent classiﬁcation using types of probability as an example.
Each day on his way to work, Steve drives up to thebusy four-way intersection in Omaha. When the traﬃc light signals green, Steve drives through the intersection. When the traﬃc light is red, he stops and waits for it to turn green. When the traﬃc light is yellow, Steve considers whether or not he will make it through the intersection in time before making a decision as to whether to stop or go. It doesn’t take a whole lot of eﬀort for Steve to makeit into work. Without the traﬃc light in place, making it through the intersection might be a chore and it might not even be possible. He would always have to slow down, prepared to stop if necessary. He’d need to look around to see if there are cars coming from the other three directions that might cross his path. If there are enough other drivers, the whole process would be chaos! It’s a goodthing we have traﬃc lights to make driving easier. Now, let’s build a probability traﬃc light! We can classify three main types of probability problems based on the ways in which we can approach them: counting, geometry, and algebra. When we can identify these types as easily as the colors on a traﬃc light, we can cut to the chase and solve problems.
Thisarticle is excerpted from the booklet Art of Problem Solving Introductory Topics Presented for MATHCOUNTSTM at NASA Educators Workshop September 13, 2005
Probability as Counting
The ﬁrst type of probability we will discuss is perhaps the simplest to understand. Let P(event) be the probability of some event occuring. We can often determine P(event) by counting the number of successfuloutcomes and then dividing by the total number of equally likely outcomes: Concept: P(event) =
# of successful outcomes # of total outcomes
Let’s take a look at a couple of problems that apply this principle of counting to solve probability problems. Problem 1.1: Find the probability that when two standard 6-sided dice are rolled, the sum of the numbers on the top faces is 5. Solution forProblem 1.1: There are 6 · 6 = 36 possible outcomes when we roll a pair of dice. We can list the outcomes in which the sum of the top faces is 5:
¥+¨ ¦+§ §+¦ ¨+¥
P(Sum of 5) =
= = = =
5 5 5 5
We can now reach an answer by dividing the number of successful outcomes by the total number of possible outcomes: # of successful outcomes 4 1 = = . # of total outcomes 36 9
Now let’s take alook at another example of counting probability that requires a bit more thought: Problem 1.2: A bag contains 16 marbles, 4 of which are blue and 12 of which are green. Two marbles are randomly pulled from the bag at the same time. What is the probability that both marbles are blue? Solution for Problem 1.2: counting methods. We can work this problem in several ways – all of which are based inIn our ﬁrst solution we note that it doesn’t matter that both marbles are drawn at once. We can arbitrarily call one of them the ﬁrst marble and the other the second marble. There are 16 choices for 2 www.artofproblemsolving.com
This article is excerpted from the booklet Art of Problem Solving Introductory Topics Presented for MATHCOUNTSTM at NASA Educators Workshop September 13, 2005 the...