Multiplication and Probability

Problem Solving: Multiplication and Probability Problem Solving: Multiplication and Probability What is an efficient way to figure out probability? In the last lesson, we used a table to show the probability of flipping a coin to get heads and then flipping the coin a second time to get heads again. We used a tree to show the probability of flipping heads first and then tails. Each of these outcomes had a probability of 4, or 0.25. Now let s look at something a little more complicated. What are the chances of rolling 5 and 2 with a pair of dice? Throughout this unit, we reminded ourselves to think about the problem carefully. In this case, think about the possible ways we could roll a 5 and a 2. The problem does not say that one of the dice has to be a 5 and the other one has to be a 2. Either die could be a 5 or a 2. That means there are two ways to get a 5 and a 2. There is one other thing to think about. It doesn t matter how we roll the dice. We could roll the first one, wait, and then roll the second one. Or we could roll both dice at the same time. One roll doesn t have any effect on the other roll. They are independent of each other. It also doesn t matter which number comes up first. The order is not important. The problem just stated that we wanted a 5 and a 2. 58 Unit 7 Lesson 8

Example Find the probability of rolling a 2 and a 5. First Die Second Die First Die Second Die 4 2 4 2 3 4 3 4 4 4 5 4 5 6 4 6 2 5 2 2 5 2 z 2 3 5 3 2 4 5 4 2 5 z 5 5 2 6 5 6 3 6 3 2 6 2 3 3 6 3 3 4 6 4 3 5 6 5 3 6 6 6 Outcomes with 5 and 2: 2 Total possibilities: 2 = 0.055 or 5.5% (rounded to 6%) The probability is 0.055, or about 6%. Unit 7 Lesson 8 59

It should be clear that our tables are starting to get rather large. The table in Example shows all possible outcomes for rolling just two dice. What if we wanted to know, What are the chances of rolling three s? The table would get much bigger. Fortunately, mathematicians have figured out an easier way to calculate that kind of answer using multiplication. Let s start by asking, What are the chances of rolling two s? Think about how we set up the problem. The question is asking, What are the chances of rolling a and a? This question means that we want both a and a. Multiplication helps us find the probability of more than one event occurring. The chances of rolling a are 6. Let s say we roll the first die, and it is a. What are the chances of rolling a with the next die? They are still 6. The roll of the first die did not have any effect on what is going to happen with the second die. Example 2 shows how we calculate the probability of rolling a and a. It also shows how we calculate rolling three s. Example 2 Calculate the probabilities for rolling s. The chances of rolling two s: 6 6 = The probability is = 0.028, or about 3%. The chances of rolling three s: 6 6 6 = 26 The probability is 2 6 = 0.0046, or about 0.5%. 520 Unit 7 Lesson 8

We don t always have to pick the same number to think about probabilities in this way. Suppose that we forgot our combination to a bicycle lock. There are four rings on the lock. We just have to line up the right combination to open it. Each ring has 0 numbers 0 through 9. We might think, This won t take long. I ll just guess and see if I can figure it out. Example 3 shows that the chances of guessing the right combination are very small. Example 3 Find the probability of figuring out the combination. Probability of the right combination number for: The first ring: 0 The second ring: 0 The third ring: 0 The fourth ring: 0 The probability of getting all 4 numbers correct is 0 0 0 0 = 0,000, or 0.000% This is why there are so many numbers on combination locks. It would take someone a long time just to guess the right numbers in the combination. Problem-Solving Activity Turn to Interactive Text, page 27. Reinforce Understanding Use the mbook Study Guide to review lesson concepts. Unit 7 Lesson 8 52

Homework Activity Rewrite each number using scientific notation.. 0.00000000 2. 3,400,000 3. 0.000000045 4. 48,000,000 5. 0.00023 6. 4,570,000 Activity 2 Multiply the probabilities to find the combined probability of each event. You may use a calculator.. The probability of rolling three 4s is 6 6 6. What is the probability? 2. The probability of figuring out the combination of a 5-ring lock is 0 0 0 0 0. What is the probability? 3. The probability of getting heads on 5 different coins is 2 2 2 2 2. What is the probability? 4. The probability of selecting a heart out of four separate decks of cards is 4 4 4 4. What is the probability? Activity 3 Answer true or false to each question about rolling five dice.. You have a better chance of getting five s than five 6s. 2. You have a better chance of getting three 4s than four 3s. 3. The chance of rolling three 3s and two 4s is 6 6 6 or 26. 4. You have a better chance of getting five 4s than four 5s. Activity 4 Distributed Practice Solve. 26 00,000 32 256. Write 25% as a fraction. 4 2. Write a decimal number and a percent for 0. 3. Convert 0.002 to a percent. 4. 8.8 0.2 5. 7 2 2 7 6. 5.006 3.789 522 Unit 7 Lesson 8 Copyright 200 by Cambium Learning Sopris West. All rights reserved. Permission is granted to reproduce this page for student use.