## Lesson: Finding Probability Introducing the Concept

Your students may have some intuitive notions about probability. Discuss with them the idea of an event being certain, likely, equally likely, unlikely, or impossible. Ask them to identify events that could be classified as each of these categories.

Materials: a bag with 10 tiles numbered 1 to 10 (You could use cubes or some other objects instead of tiles. The objects must be identical except for the numbers written on them.)

• Say: I have a bag and 10 tiles numbered 1 to 10.
Show students the materials and put them in the bag.
• Ask: If I reach into the bag without looking and pick out one tile, what number do you think it will have on it?
Students will probably say that it could have any of the numbers, since they are all equally likely.
• Ask: Do you think we are more likely to pick a number less than 3 or greater than 3?
Students should say that it is more likely that the number will be greater than 3, since there are only 2 tiles less than 3, but there are 7 tiles greater than 3.
• Ask: There is a field of mathematics that investigates whether an event is likely to happen or not. Do you know what that field is called? (probability)
• Say: The probability of an event ranges from zero to one. An event that is sure to happen has a probability of one, and an event that cannot happen has a probability of zero. The probability that I could reach in this bag and draw out an elephant is zero. The probability that tomorrow will come is 1.
• Say: The result of an experiment like this (reaching in the bag and pulling out a numbered tile) is called an outcome. We mentioned before that each numbered tile had an equal chance of being drawn. Since the tiles have an equal chance of being selected, the outcomes are said to be equally likely.
• Say: One or more outcomes of an experiment make up what we call an event. When the outcomes are equally likely, the probability of an event is the number of successful outcomes for the event divided by the total number of outcomes.
Write on the board the following equation.
• Say: Let's look at the event of reaching into the bag without looking and drawing a tile with an even number on it. How many even-numbered tiles are there? (5)
How many tiles are in the bag? (10) So the probability of reaching into the bag and drawing an even-numbered tile is , or .
• Say: One device that can help us find the probability of an event is a sample space. A sample space lists all the possible outcomes, making it easier to find the outcomes that are make up the event.
Write the numbers 1 to 10 on the board.
• Say: The numbers 1 to 10 are a sample space for selecting a numbered tile from the bag, since they list all the possible outcomes. Let's look at some more events.
• Ask: What is the probability of selecting a tile from the bag with a number from 1 through 10? Explain.
Students should say that the probability is 1, since the event includes all possible outcomes.
• Ask: What is the probability of selecting a tile from the bag that will be greater than 6? Explain.
Students will probably say that there are 4 numbers greater than 6—7, 8, 9, and 10—so the probability is , or .
• Ask: What is the probability of drawing a tile from the bag that is a multiple of three?
Since there are three multiples of 3, namely 3, 6, and 9, the probability of selecting a multiple of 3 is .
• Ask: Let's try another one. What is the probability of picking a tile with a prime number on it?
Students should say that the prime numbers less than 10 are 2, 3, 5, and 7. Therefore, the probability would be , or .
• Ask: What is the probability that the tile will have the number 17 on it? (Students may be baffled at first, but will probably tell you that you can't get the number 17 because no tile has that number, so the probability of that happening is zero.)
Try a few more simple events like this to be sure that students understand how to find the probability of these kinds of events. Then go on to the next problem.
• Say: Let's consider the experiment of flipping a coin three times. The first thing we need to do is to figure out how many different outcomes are possible. One way to figure that out is to use the fundamental counting principle.
• Ask: Who can tell us what the fundamental counting principle says? (If an experiment or a problem has two steps and the first step can be done in m different ways and the second step can be done in n different ways, then the experiment can be done in m x n ways.)
• Say: If I flip a coin, it can land on either heads or tails. Thus, when flipping a coin three times, there are 2 ways to do the first flip, 2 ways to do the second flip, and 2 ways to do the third flip for a total of 2 x 2 x 2, or 8, possible outcomes. You can find all 8 outcomes by using a tree diagram.
On the board, draw a tree diagram like the one below and show students how to find the 8 possible outcomes by tracing each branch from left to right.
• Ask: What is the probability that all three flips will be heads? Students will probably say that there is only one result which is all heads—HHH—so the probability is .
• Ask: What is the probability that there are exactly 2 heads and 1 tail in the three flips?
Students will probably say that there are three outcomes with exactly 2 heads and one tail, so the probability is .

Ask a few more questions like the ones above to be sure your students understand how to use a sample space to find the probability of an event.