## Lesson: Finding Probability Developing the Concept

Now that students have had an opportunity to find the probability of simple events by using a sample space or the fundamental counting principle, they can apply these strategies on their own.

Materials: 5 red, 3 green, and 2 yellow tiles; a paper bag; paper and pencil for each student

Prerequisite Skills and Background: Students should be able to apply the fundamental counting principle in a problem-solving situation.

On the board, write “5 red,” “3 green,” and “2 yellow” to indicate the color of the tiles in the bag.

• Say: Today we are going to look at finding the probability of events by drawing tiles from a bag. I have placed 5 red, 3 green, and 2 yellow tiles in the bag. If I reach into the bag and draw a tile without looking, what is the probability it will be green? Students will probably respond that the probability is .
• Ask: That's right. What is the probability of drawing a yellow tile? (, or )
What is the probability of drawing a red tile? (, or )
• Say: That's good. Now for a more difficult problem. What is the probability of reaching in and not drawing a yellow tile? Students may struggle with this, but will probably say that there are 8 tiles that are not yellow, so the probability will be , or . If they don't say this, you might ask them how many tiles there are and how many are not yellow.
• Ask: What is the probability of drawing a blue tile? (zero, since there are no blue tiles in the bag)
• Ask: What is the probability of reaching into the bag and not drawing a purple tile? (Since there are no purple tiles, the probability would be 1.)
• Say: Now we are going to investigate events that are a little more complex. Let's think about reaching into the bag and drawing out a tile, recording its color, replacing the tile in the bag, and then reaching in to get a second tile.
Do this so students can see exactly what you are saying.
• Say: The question I would like us to solve is, “What is the probability that both tiles drawn will be red?” How might we solve this problem? (Students may suggest using the fundamental counting principle.)
• Say: The first thing to do is to find out how many ways I could select two red tiles by reaching in and drawing a tile, recording it, replacing it, and then selecting a second tile.
Students will say that you can select any of 5 red tiles each time you draw, since you replace the tile.
• Say: Since I can draw any of 5 tiles on the first draw and any of 5 tiles on the second draw, these are independent events. So I can draw two red tiles 5 x 5, or 25, ways according to the fundamental counting principle.
• Ask: So how do we find the probability of these two independent events? If I reach into the bag and draw a tile, how many different outcomes are possible? (10) If I replace that tile and reach in and draw again, how many different outcomes are possible? (10)
• Say: Good, so there are 10 x 10, or 100, ways I could draw 2 tiles from the bag by replacing the tile after I draw the first time. Therefore, the probability of drawing two red tiles is , or .
• Say: Now we are going to change the problem slightly. Instead of replacing the tile, we keep it and draw a second tile. Now the two events are dependent events. How many different ways can these events occur?
Students should say that the first draw can occur 10 different ways and the second draw, 9 different ways; therefore, you can draw twice without replacing the tile in 9 x 10, or 90, ways.
• Ask: Keeping in mind that we are not replacing the tile, what is the probability of drawing 2 red tiles?
Give students time to think about it, and then ask for a volunteer to explain what he or she did. There would be 5 x 4, or 20, ways of selecting two red tiles out of 90 ways, if the first tile drawn is red, so the probability would be or .
• Ask: Again, if we don't replace the tile, what is the probability of drawing a green tile followed by a red tile?
Students will probably say that there are 3 ways to draw a green tile and 5 ways to draw a red tile, so there are 15 ways to draw a green tile followed by a red tile. Therefore, the probability would be , or .
• Say: You can see how important it is to know whether or not you replace the tile when drawing more than one tile.
Continue asking similar questions. If students seem to be catching on, you can go to selecting three tiles with or without replacing them.

Wrap-Up and Assessment Hints
“What if” questions are good ways to assess the depth of a student's understanding of probability. Most problems, like those you have done, have several factors that could affect the outcome. In those problems, the number of tiles of each color, the number of draws, and drawing with or without replacement are variables that you can alter to assess the depth of students' understanding. Drawing two tiles with replacement is an example of two independent events. The outcome of the first event does not affect the second event. Drawing tiles without replacing them is an example of dependent events. These are concepts your students should become familiar with.