1. Review of Probability Grade 6 Copyright © Ed2Net Learning Inc.1

2. 2 Warm Up 1.Find the theoretical probability of rolling an even number with a die Suppose you have a bag containing 2 red marbles,2 blue marbles, and 2 white marbles. You can choose a marble without looking. 2. What is the theoretical probability that you will choose a white or a blue marble?

3. Copyright © Ed2Net Learning Inc.3 4. Find the theoretical probability of choosing a girl’s name at random from 20 boy’s names and 10 girl’s names. 5. Find the theoretical probability that a family of 4 children will be all girls. 3. Find the theoretical probability of choosing a winning 3-digit number in the lottery. Warm Up

4. Tree Diagram There are two basic types of trees. Unordered Tree Ordered Tree Copyright © Ed2Net Learning Inc.4 In an unordered tree, a tree is a tree in a purely structural sense A tree on which an order is imposed — ordered Tree A node may contain a value or a condition or represents a separate data structure or a tree of its own. Each node in a tree has zero or more child nodes, which are below it in the tree

5. Copyright © Ed2Net Learning Inc.5 A Sub tree is a portion of a tree data structure that can be viewed as a complete tree in itself A Forest is an ordered set of ordered trees Traversal of Trees In order Preorder Post order In graph theory, a tree is a connected acyclic graph.

6. Copyright © Ed2Net Learning Inc.6 Preorder And Post order Walk A walk in which each parent node is traversed before its children is called a pre-order walk; A walk in which the children are traversed before their respective parents are traversed is called a post-order walk.

7. Copyright © Ed2Net Learning Inc.7 Tree diagram The probability of any outcome in the sample space is the product (multiply) of all possibilities along the path that represents that outcome on the tree diagram. A probability tree diagram shows all the possible events. Example: A family has three children. How many outcomes are in the sample space that indicates the sex of the children?

8. Copyright © Ed2Net Learning Inc.8 There are 8 outcomes in the sample space. The probability of each outcome is 1/2 1/2 1/2 = 1/8. Assume that the probability of male (M) and the probability of female (F) are each 1/2.

9. Counting Principle Event : An Event is an experiment. Outcome: Possible outcomes of an event are the results which may occur from any event. Copyright © Ed2Net Learning Inc.9 Counting principle: Counting principles describe the total number of possibilities or choices for certain selections.

10. Independent & Dependent Events Dependent events Two events, A and B, are dependent if the fact that A occurs does not affects the probability of B occurring. Examples: Choosing a ball from a box AND choosing another ball from that box. If A and B are dependent events, the probability of both events occurring is the product of the probability of the first event and the probability of the second event once the first event has occurred. P(A and B) = P(A) x P(B, once A has occurred ) Copyright © Ed2Net Learning Inc.10

11. Copyright © Ed2Net Learning Inc.11 Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring. If A and B are independent events, the probability of both events occurring is the product of the probabilities of the individual events P(A and B) = P(A) x P(B). Independent events Choosing a ball from a box AND landing on tails after tossing a coin. Examples:

12. Combinations 1. The fundamental counting principle says that, the the total number of different ways a task can occur is: n1 * n2 * n3 * ………………… * nr where n1, n2, n3..nr are the tasks performed at first, second and rth stages 2. Factorial is a shorthand notation for a multiplication process. The Fundamental Counting Principle says that the number of ways to choose n items is n ! n! = n * (n-1) * (n-2) *……………*3 * 2 * 1 3. Any factorial less than n! is a factor of n! Copyright © Ed2Net Learning Inc.12

13. Copyright © Ed2Net Learning Inc.13 5. The number of ways (or combinations) in which r objects can be selected from a set of n objects, where repetition is not allowed, is denoted by, or C(n, r) n C r n! n C r = (n-r)! r! 6. Pascal’s Triangle illustrates the symmetric nature of a combination. i,e C(n,r) = C(n,n-r) 4. A combination focuses on the selection of objects without regard to the order in which they are selected

14. Copyright © Ed2Net Learning Inc.14 10. The sum of all combinations : C(n,0) + C(n, 1) +... + C(n, n) = 2^n 11. Relationship between permutation and combination is p n r n C r = r! 7. C(n, 1) = n 8. C(n, 0) = 1 9. C(n, n) = 1

15. Permutations Copyright © Ed2Net Learning Inc.15 The symbol n P r is used to indicate the number of permutations n objects taken r at a time. n P r = n!/ (n - r)! The numbers of n objects is n!. If a set of n elements has n 1 elements of one kind, n 2 of another kind alike and so on, then the number of permutations, P, of the n elements taken n at a time is given by: P = n! n 1 !.n 2 !.......

16. Experimental Probability 1.Probability is the ratio of the different number of ways a trial can succeed (or fail) to the total number of ways in which it may result 2.Probability mathematically lies between two limits 0 and 1 3.An experiment is a method by which observations are made. 4.A possible result of a probability experiment is called an outcome. 5.The sample space is the set of all possible outcomes for a given experiment. Each possible result of such a study is represented by one and only one point in the sample space, which is usually denoted by S Copyright © Ed2Net Learning Inc.16

17. Copyright © Ed2Net Learning Inc.17 6. An event(E) is a one or more outcome of an experiment. 7. A set of outcomes of an event are said to be equally likely if they all have the same choice of happening. 8. The probability of success (p) is P = s / (s + f) 9. The probability of failure (q) is, q = f / (s + f) 10. Relative frequency is the value calculated by dividing the number of times an event occurs by the total number of times an experiment is carried out. 11. The probability of the event is defined as the limiting value of the relative frequency.

18. Copyright © Ed2Net Learning Inc.18 12. Experimental Probability or Estimated Probability is the ratio of the number of times an event occurred to the number of times tested. P = fr(E)/N 13. The numerical expectation of an event is the product of the probability of success in one trial and the total number of trials. En = kp 14. The mathematical expectation of an event is the product of amount to be received and the probability of success of that event.Em = ap

19. Theoretical Probability 1. Probability is a numerical measure of the likelihood of occurrence of an event. 2. An experiment is a situation involving chance or probability that leads to results called outcomes. 3. An event is one or more outcomes of an experiment. 4. The sample space is the set of all possible outcomes for a given experiment. 5. A probability will be expressed as a fraction, a decimal, or a percent. 6. The probability(P) of any event A is a number between zero and one. Copyright © Ed2Net Learning Inc.19

20. Copyright © Ed2Net Learning Inc.20 7. The sum of the probabilities of all the outcomes in the sample space is one. 8. A set of outcomes of an event are said to be equally likely if they all have the same choice of happening. 9. If all the outcomes in a sample space are equally likely, and E is an event within that sample space, then the theoretical probability of the event E is the ratio of number of favorable outcomes to the total number of outcomes. 10. The probability that an event E will occur is one minus the probability that it will not occur.

21. Copyright © Ed2Net Learning Inc.21 11. 0 ≤ P(E) ≤ 1 (The probability of an event is a number from 0 through 1, inclusive.) 12. P( ∅ ) = 0 (The probability of an impossible event is 0.) 13. P(S) = 1 (The probability of a certain event is 1.) 14. The probability of an event is the “long run” relative Frequency. 15. Two events are said to be independent if they cannot influence or affect each other. 16. Two events are said to be dependent if the occurrence or non occurrence of one of the events affects the probabilities of occurrence of any of the others.

22. Copyright © Ed2Net Learning Inc.22 17. Outcomes are said to be mutually exclusive if any of those outcomes prevents the other from happening. 18. The AND rule: p (A  B) = p(A)  p(B) 19. The OR rule: P(A  B) = p(A) + p(B)

23. Copyright © Ed2Net Learning Inc.23 Break Time

24. Copyright © Ed2Net Learning Inc.24 Your Turn Suppose you have a child’s play cube with one of the following letters on each face: A,B, C, D, E, F. You toss the cube 1.What is the theoretical probability of turning up an A, B, or C? Lara tosses two coins four times. Twice both the coins come up with heads. 2. What is the theoretical probability of getting two heads? 3. For the given situation tell the number of outcomes. Rolling a number cube, tossing a coin, and choosing a card from among the cards marked W, X, Y, and Z.

25. Copyright © Ed2Net Learning Inc.25 Your Turn 4. In a dance notation, movements are described as either light or strong in the weight factor, sustained or quick in the time factor, and direct or flexible in the space factor. How many weight-time-space combinations are possible? Use counting principle to find the total number of outcomes in the case given 5. Choosing one of three science courses, one of 5 math's courses, one of 2 English courses, and one of 4 social studies courses.

26. Copyright © Ed2Net Learning Inc.26 Your Turn 6. Daniel has a package of fruit candy containing two pieces each of raspberry, orange, lemon, and cherry flavors. Find the probability that he chooses both lemon pieces first. Tell whether the event is independent or dependent. 7. Selecting a compact disc from a storage case and then selecting a second disc without replacing the first. 8.List all the permutations of the digits 2, 4, and 6.

27. Copyright © Ed2Net Learning Inc.27 9. Tell whether there are more combinations of the objects taken three at a time or permutations of a set of objects taken three at a time. 10. Explain why a combination lock really should be called a permutation lock. Your Turn

28. Copyright © Ed2Net Learning Inc.28 1. Jack has 18 black balls and 10 white balls. Find the probability that he chooses first black ball and then he chooses white ball.

29. Copyright © Ed2Net Learning Inc.29 2. In how many ways can a coach choose first a football team and then a basketball team from 18 boys?

30. Copyright © Ed2Net Learning Inc.30 3. Dennis rolled 2 number dice 72 times. He recorded the results in the table as shown. Find the experimental probability of rolling an even sum.

31. Copyright © Ed2Net Learning Inc.31