If event A is more likely than event B, then the probability of event A is higher than the probability of event B.

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1 Unit, Lesson. Making Decisions Probabilities have a wide range of applications, including determining whether a situation is fair or not. A situation is fair if each outcome is equally likely. In this section, you will evaluate whether a given situation is fair based on the probabilities of the outcomes. You will also assess whether a random number generator will truly produce a random result based on probabilities. The likelihood of events occurring can be given as a probability. ts that are equally likely have the same probability. If event A is more likely than event B, then the probability of event A is higher than the probability of event B. If a situation is fair, each outcome is equally likely. If a game is fair, each player has the same probability of winning. To maintain fairness, random numbers are often used in making selections. A random number generator is a tool to select a number without following a pattern such that each number in the set has an equal chance of being selected. A random number generator selects one number from a set of numbers. The set can be small or large, but there is a smallest and largest value. A random number generator does not select numbers using a repeated pattern, and the probability of selecting one number is not greater than the probability of selecting any other number. To determine whether a random number generator is selecting numbers randomly, take a look at how the numbers are being selected. If the numbers are being selected in a pattern or if it is more likely that certain numbers will be selected, then the generator is not random. The results from a random number generator do not always reveal whether the generator is random. For example, if the results of a random number generator are 5, 2, 2, 7, 9, it may appear that selecting a 2 is more likely than selecting other numbers. This may or may not be true. Always take a closer look at how the numbers are selected, instead of which numbers are selected. Common Errors/Misconceptions incorrectly calculating probabilities using a list of randomly generated numbers to determine if the generator is random instead of examining the generation method UnitLesson. 0/2/207

2 Unit, Lesson. Making Decisions (continued) Example : Jose and Cash are playing a game with 2 dice. Each die has 6 sides, numbered through 6. Jose earns a point if either of the dice is even. Cash earns a point if both dice are odd. Is the game fair?. Find all the possible outcomes for the game. There are 2 dice. The dice can either be even or odd. The different ways the dice can be rolled are: First die There are possible outcomes. 2. Determine the probability of each outcome. Rolling one die does not influence the roll of the other, so the two rolls are independent. The probability of the two events occurring is the product of the probabilities of each event. The number of w aysto roll an odd number probability that any die is odd is. There are 6 sides on number of w aysthe die can be rolled each die, numbered, 2, 3,, 5, and 6. Of these, 3 are odd numbers:, 3, and 5. The number of w aysto roll an odd number probability that a roll is odd is as follows: P( ) number of w aysthe die can be rolled Similarly, there are 3 even numbers on each die: 2,, and 6. The probability that a roll is even number of w aysto roll an even number is as follows: P( ) number of w aysthe die can be rolled Second die Use these probabilities to find the probabilities of rolling each of the possible combinations. First die Second die Probability UnitLesson. 2 0/2/207

3 Unit, Lesson. Making Decisions (continued) Example : (continued) 3. Find the probability that each player earns a point. The table can be used to help find the probabilities. Determine who will win each type of roll. First die Second die Probability Winner Jose Jose Jose Cash The probability that Jose will win is the probability that any of the first three pairs are rolled. He can roll even even, even odd, or odd even. The probability that he will win is: P(even even or even odd or odd even). Since each pair is independent, his probability of winning is P(both even) + P(one even and one odd) + P(one odd and one even) The probability that Cash will win is the probability that the last pair, odd odd, is rolled. His probability of winning is P(both odd ) Use the probabilities to determine if the game is fair. The probability that Jose will earn a point is 0.75, and the probability that Cash will earn a point is Because the probability that Jose will earn a point is greater than Cash s probability, it is more likely that Jose will earn a point. Because it is more likely Jose will earn a point, the game is not fair. UnitLesson. 3 0/2/207

4 Unit, Lesson. Making Decisions (continued) Example 2: Morgan creates a random number generator to help her pick a winning combination of numbers. There are 2 winning numbers. The available numbers range from through 5. To select the first number, she writes each number on a different piece of paper, puts the 5 pieces of paper in a hat, and picks one without looking. To select the second number, Morgan uses the table she created below, so that her second number is based on the first number she draws from the hat. First Number Second Number Is Morgan s selection of the 2 winning numbers random?. Determine if selecting the first number follows a pattern. Morgan selects the numbers from a hat, without looking. This does not follow a pattern. 2. Determine if selecting any given value of the first number is more likely. Morgan has written each of the 5 different numbers on a different piece of paper. The probability that any of the 5 numbers are selected is number of times a number is in the hat. Each number has the same probability of number of numbers in the hat 5 being selected, so there are no values that are more likely. 3. Identify whether the generator is random for the first winning number. Because Morgan is selecting the first number without following a pattern and the probability of selecting each number is the same, the generator is randomly selecting the first winning number. UnitLesson. 0/2/207

5 Unit, Lesson. Making Decisions (continued) Example 2: (continued). Determine if selecting the second number follows a pattern. The second number is selected base on the value of the first number, using the table below. First Number Second Number For example, every time she picks 0 as the first number, the second winning number will be ; every time she picks 3 as the first number, the second winning number will be 9, and so on. Matching each first number with only one second number is a pattern. The second number is not random; it is based on the result of the first number. 5. Determine if selecting any given value of the second number is more likely. The values in the table show that each number, through 5, is only listed once as a winning number. Because the second number depends on the first, the probability that any of the second numbers will win is equal to the probability that the first number is selected. For example, the probability that 8 is the second number is the probability that 5 is the first number, since 5 and 8 are always chosen together as the first and second winning numbers. The probabilities of selecting each of the first numbers are equal, and each second number is different, so the probabilities of selecting each second number are also equal. None of the second numbers are more likely. 6. Identify whether the generator is random for the second winning number. The selection of the second winning number, even though each value is equally likely, follows a set pattern. Therefore, selecting the second winning number is not random. 7. Identify whether the generator is random for selecting a winning pair of numbers. Because selecting the second number is not random, Morgan s random number generator is not random for selecting a winning pair of numbers. UnitLesson. 5 0/2/207

6 Second die Integrated Math II Honors Mathematics II, Walch Integrated Math Unit, Lesson. Making Decisions (continued) Example 3 Mr. Won is taking his class on a field trip. He wants to find a random way to choose a student to be the class leader on the trip. There are students in his class. He assigns each student a unique number from through. He decides to use 2 dice to determine which student is selected. Each die has 6 sides, numbered though 6. The product of the dice will be the number of the student. Is his selection fair?. Find all possible outcomes of Mr. Won s number generator. Mr. Won is using the product of two dice to determine a student s number. Find the possible products of the two 6-sided dice. First die Organize the results in the table so that the products are from least to greatest. Each product represents a student s assigned number. Die Die 2 Product Die Die 2 Product Die Die 2 Product Determine if selecting the student number follows a pattern. Selecting a number depends on the rolling of 2 dice. Rolling dice does not follow a particular pattern, so the number is not selected following a pattern. UnitLesson. 6 0/2/207

7 Unit, Lesson. Making Decisions (continued) Example 3 (continued) 3. Determine if any given student number from through is equally likely to be selected. The numbers through have all been assigned to different students. There are different ways Mr. Won can roll the 2 dice. Each of these different pairs is equally likely. For example, calculate the probability of rolling and a. The probability of rolling a is P() number of w aysa can be rolled number of w aysto roll the die. 6 Each roll is independent, so the probability of both events is P( and ) P() P() 6 6 The probability of each of the roll pairs can be found in the same way. The probability of rolling each pair, or each product, is equally likely. However, some of the numbers between and cannot be represented as a product of the 2 dice, and some numbers appear more than once. For example, the product 2 can occur two different ways, and the product can occur three different ways, but there is no product equal to 35. The probability of rolling a product of 2 is P(rolling then 2 or rolling 2 then ). Each pair of rolls is independent. P( then 2 or 2 then ) = P( then 2) + P(2 then ). The probability of rolling each pair is, so the probability of rolling then 2 or rolling 2 then is P( then 2) P( 2 then ) 2 8 However, the probability of rolling a product of 35 is 0, because 35 is not the product of any rolls. Given that two numbers have different probabilities, there are at least two numbers that are not equally likely. Selecting a student number from though is not equally likely.. Identify whether the method fairly selects students. While the student numbers are selected without following a pattern, the selection of each student number is not equally likely. Therefore, the method does not fairly select students. UnitLesson. 7 0/2/207

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