The document discusses key concepts in probability, including tree diagrams, sample spaces, theoretical and experimental probability, and ratios and rates. It provides examples of how to use tree diagrams to calculate probabilities of outcomes from multiple choices. It also gives examples of calculating greatest common factors, least common multiples, and solving ratio and rate problems. Interactive questions and explanations are included throughout to illustrate the concepts.

1. Unit 12: Probability

2. Key Goals Understand and use tree diagrams to solve problems Compute the probability of outcomes when choices are equally likely Use the multiplication counting principle to find the total number of possible outcomes of a sequence of choices. Find the greatest common factor and least common multiple of two numbers Solve ratio and rate number stories Find the factors and prime factorizations of numbers Click me for help Click me to move on

3. Understand and use tree diagrams to solve problems A tree diagram is a branched diagram that shows all possible choices. Tree diagrams are useful for solving probability problems when there are several options

4. Tree diagrams can be a helpful way of organizing outcomes in order to identify probabilities. For example, if we have a box with two red, two green and two white balls in it, and we choose two balls without looking, what is the probability of getting two balls of the same color? We use the tree diagram to the left to help us identify the possible combinations of outcomes. Here we see that there are nine possible outcomes , listed to the right of the tree diagram. This number is the size of the sample space for this two step experiment, and will be in the denominator of each of our probabilities. Each of these possible nine outcomes has a probability of 1/9. Because there are three instances out of the nine that result in 2 balls of the same color: P(2 the same color)= 3/9 (This can be reduced to 1/3)

5. Try it out! What is the probability of getting a white ball in the second stage? 4/9 1/3 2/3

6. Try it out! What is the probability of getting a white ball in the second stage? The probability is 1/3 because of the 9 possibilities, there are 3 white balls. 3/9 = 1/3 4/9 1/3 2/3

7. Try it out! What is the probability of getting a white and a green ball in no particular order? 2/9 1/3 2/3

8. Try it out! What is the probability of getting a white ball in the second stage? The probability is 2/9 because of the 9 possibilities, there are 2 combinations with those colors- WG and GW If the order did matter, then the probability would be 1/9 because there is a 1/3 chance of the first ball’s color being white and a 1/3 chance of the second ball being green. 1/3 x 1/3 = 1/9 4/9 1/3 2/3

9. Compute the probability of outcomes when choices are equally likely. Probability can be determined by giving the total number of desired outcomes over the total number of possibilities. For example, on a six-sided die, there is a 1/6 chance of rolling a 4. If you wanted to find the chances of rolling a 2 or a 4, you would add the probability of getting a 2 to the probability of getting a 4. If you wanted to know the probability of getting a 2 and then a 4 ( order matters ), then you would multiply the probabilities together. 1/6 x 1/6 = 1/12

10. Try it out! Read about probability and take a quiz Play probability games Crazy choices game Theoretical vs. Experimental probability

11. Use the multiplication counting principle to find the total number of possible outcomes of a sequence of choices. When you have a sequence events, the likelihood of the events happening together in a particular order is less than the probability of them happening separately or in no particular order. When you have dependent events, you will multiply the probabilities together to calculate the probability of both events happening. - You do this because probability is expressed as a fraction. If you add two fractions together, the sum will be greater than each fraction. When you multiply two fractions together, the product is smaller than the first fraction because you are finding a part of a part.

12. Try It Out! Click here to print a probability activity. Use the dice and the coins on this page to help answer the questions. Turn your work in to Mrs. R. when you finish.

13. Name _______________ Use the links on the previous slide to help you complete this chart. Give the theoretical and experimental probability for each event described and analyze and explain the relationship between your theoretical and experimental probability. Click me to return to the previous slide Remember that theoretical probability is based only on math. Experimental probability is based on actual experimenting. Do your theoretical and experimental probabilities vary greatly? Why or why not? Experimental probability Theoretical probability Describe events

14. Find the Greatest Common Factor and Least Common Multiple of two numbers The Greatest Common Factor (GCF) is the largest number that two numbers are both divisible by. To find the GCF of two numbers: List the prime factors of each number. Multiply those factors both numbers have in common. If there are no common prime factors, the GCF is 1. Explanation and Game

15. Find the Greatest Common Factor and Least Common Multiple of two numbers The Least Common Multiple (LCM) is the smallest multiple that both numbers have in common. A common multiple is a number that is a multiple of two or more numbers. The common multiples of 3 and 4 are 0, 12, 24, .... The least common multiple (LCM) of two numbers is the smallest number (not zero) that is a multiple of both. Explanation and Game

16. Try It Out! Which is the LCM of 6 and 10? 2 60 20 30

17. Try It Out! Which is the LCM of 6 and 10? The LCM of 6 and 10 is 30 If you list the multiples of 6, you have: 6, 12, 18, 24, 30 If you count by 10s, the first one of those numbers you come to is 30. 2 60 20 30

18. Try It Out! Which is the LCM of 12 and 8? 12 60 24 30

19. Try It Out! Which is the LCM of 12 and 8? The LCM of 12 and 8 is 24 If you list the multiples of 8, you have: 8, 16, 24, 32 If you list the multiples of 12, the first of those numbers you come to is 24. 12 60 24 30

20. Try It Out! Which is the GCF of 12 and 8? 12 8 4 24

21. Try It Out! Which is the GCF of 12 and 8? The GCF of 12 and 8 is 4 If you list the factors of 8, you have: 1, 2, 4 , 8 If you list the factors of 12, you have: 1, 2, 3, 4 , 6, 12 4 is the largest factor they have in common. 12 8 4 24

22. Try It Out! Which is the GCF of 24 and 38? 12 2 4 24

23. Try It Out! Which is the GCF of 24 and 38? The GCF of 24 and 38 is 2 If you list the factors of 38, you have: 1, 2, 19, 38 If you list the factors of 24, you have: 1, 2 , 3, 4, 6, 8, 12, 24 2 is the largest factor they have in common. 12 2 4 24

24. Solve Ratio and Rate Number Stories A ratio is a way of comparing amounts of something. It shows how much bigger one thing is than another. For example: The ratio of footballs to soccer balls is 4:3 This can also be written as 4 to 3 or 4/3

25. A rate is a ratio that compares two different kinds of numbers, such as miles per hour or dollars per pound. A unit rate compares a quantity to its unit of measure. A unit price is a rate comparing the price of an item to its unit of measure. The rate "miles per hour" gives distance traveled per unit of time. Problems using this type of rate can be solved using a proportion , or a formula .

26. Rate is a very important type of ratio, used in many everyday problems, such as grocery shopping, traveling, medicine--in fact, almost every activity involves some type of rate. Miles per hour or feet per second are both rates of speed. Number of heartbeats per minute is called "heart rate ." If you ask a babysitter, "What is your rate ?", you are asking how many dollars per hour you will be charged. The little word " per " is always a clue that you are dealing with a rate . Unit price is a particular rate that compares a price to some unit of measure. For example, suppose eggs are on sale for $.72 per dozen. The unit price is $.72 divided by 12, or 6 cents per egg. The word "per" can be replaced by the "/" in problems, so 6 cents per egg can also be written 6 cents/egg. Click me to try some practice problems!

27. Try It Out! 13 26 2 15 The Littleville basketball team tries 2 field goals for every free throw. They make half of their attempts of either try. If they make 13 free throws in a game, how many field goals are they expected to make?

28. Try It Out! 13 The correct answer is 13 because: For every free throw, they attempt two field goals. If they make 13 free throws, they would attempt 26 field goals. If they attempt 26 field goals, they will make half of them for a total of 13 expected baskets. The Littleville basketball team tries 2 field goals for every free throw. They make half of their attempts of either try. If they make 13 free throws in a game, how many field goals are they expected to make? 26 2 15

29. Try It Out! 28 18 7 11 Aaron has a record of winning 4 boxing matches for every 7 he loses. If he had 50 matches during the last year, how many did he win?

30. Try It Out! 18 The correct answer is 18 because: 4 wins 7 losses total matches= 11 4 wins 7 losses total matches= 22 4 wins 7 losses total matches = 33 4 wins 7 losses total matches= 44 2 wins 4 losses total matches = 50 18 total wins Aaron has a record of winning 4 boxing matches for every 7 he loses. If he had 50 matches during the last year, how many did he win? 28 7 11

31. Try It Out! If the tortoise moves 3 feet per minute, and the hare moves 9 feet per minute, how long will it take each to reach the finish line? 27 feet 3 min. 9 min. 3 min. 9 min.

32. Try It Out! If the tortoise moves 3 feet per minute, and the hare moves 9 feet per minute, how long will it take each to reach the finish line? 27 feet 27 feet / 3 feet = 9 minutes 27 feet / 9 feet = 3 minutes 3 min. 9 min. 3 min. 9 min.