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This article is excerpted from the booklet Art of Problem Solving Introductory Topics Presented for MATHCOUNTSTM at NASA Educators Workshop September 13, 2005

CHAPTER

1
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.

1.1

Introduction

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.

1

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Thisarticle is excerpted from the booklet Art of Problem Solving Introductory Topics Presented for MATHCOUNTSTM at NASA Educators Workshop September 13, 2005

1.2

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) =

N

# 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...