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# Probability

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### Probability

1. 1. OutlineAddition and Multiplication Rules for Probability Lecture 10, STAT 2246 Julien Dompierre D´partement de math´matiques et d’informatique e e Universit´ Laurentienne e 30 janvier 2007, Sudbury Julien Dompierre 1
2. 2. Addition Rules Outline Multiplication RulesOutline 1 Addition and Multiplication Rules for Probability Addition Rules Multiplication Rules Julien Dompierre 2
3. 3. Addition Rules Outline Multiplication RulesOutline 1 Addition and Multiplication Rules for Probability Addition Rules Multiplication Rules Julien Dompierre 3
4. 4. Addition Rules Outline Multiplication RulesMutually Exclusive Events (p. 195) Two events of the same experiment are mutually exclusive events if they cannot occur at the same time (i.e., they have no outcomes in common). U A B In this case, the intersection of the sets A and B is empty. Julien Dompierre 4
5. 5. Addition Rules Outline Multiplication RulesMutually Exclusive Events If A and B are mutually exclusive events of the same experiment, then the probability that A and B will occur is n(A ∩ B) 0 P(A ∩ B) = = = 0. n(S) n(S) For example. The experiment is to roll a die. The sample space is the set of all possible outcomes is S = {1, 2, 3, 4, 5, 6}. The event A is to get an odd number, A = {1, 3, 5} ⊆ S. The event B is to get a 6, B = {6} ⊆ S. In probability theory, we say that the events A and B are mutually exclusive because they have no outcomes in common. In set theory, we say that the sets A and B are mutually exclusive because their intersection is empty. n(A ∩ B) n(∅) 0 P(A ∩ B) = = = = 0. n(S) n(S) 6 Julien Dompierre 5
6. 6. Addition Rules Outline Multiplication RulesAddition Rule 1 (p. 196) When two events A and B of the same experiment are mutually exclusive, the probability that A or B will occur is n(A ∪ B) n(A) n(B) P(A ∪ B) = = + = P(A) + P(B). n(S) n(S) n(S) For example. The experiment is to roll a die. The sample space is the set of all possible outcomes is S = {1, 2, 3, 4, 5, 6}. The event A is to get an odd number, A = {1, 3, 5} ⊆ S. The event B is to get a 6, B = {6} ⊆ S. n(A ∪ B) n(A) n(B) 3 1 P(A ∪ B) = = + = + = 4/6. n(S) n(S) n(S) 6 6 Julien Dompierre 6
7. 7. Addition Rules Outline Multiplication RulesPrinciple of Inclusion-Exclusion (p. 197) When two events are not mutually exclusive, we must subtract one of the two probabilities of the outcomes that are common to both events, since they have been counted twice. n(A ∪ B) = n(A) + n(B) − n(A ∩ B) A B A∩B Julien Dompierre 7
8. 8. Addition Rules Outline Multiplication RulesAddition Rule 2 (p. 197) When two events A and B of the same experiment are not mutually exclusive, the probability that A or B will occur is n(A ∪ B) n(A) + n(B) − n(A ∩ B) P(A ∪ B) = = n(S) n(S) = P(A) + P(B) − P(A ∩ B). Note: This rule can also be used when the events are mutually exclusive, since (A ∩ B) will always equal 0. However, it is important to make a distinction between the two situations. Julien Dompierre 8
9. 9. Addition Rules Outline Multiplication RulesExample of Addition Rule 2 For example. The experiment is to roll a die. The sample space is the set of all possible outcomes is S = {1, 2, 3, 4, 5, 6}. The event A is to get an odd number, A = {1, 3, 5} ⊆ S. The event B is to get a number greater than 4, B = {5, 6} ⊆ S. As A ∩ B = {1, 3, 5} ∩ {5, 6} = {5} = ∅, the events A and B are not mutually exclusive. 3 2 1 4 P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = + − = 6 6 6 6 Julien Dompierre 9
10. 10. Addition Rules Outline Multiplication RulesPrincipe of Inclusion-Exclusion for Three Sets n(A ∪ B ∪ C ) = n(A) + n(B) + n(C ) − n(A ∩ B) − n(A ∩ C ) − n(B ∩ C ) + n(A ∩ B ∩ C ). A A∩B B A∩B ∩C A∩C B ∩C C Julien Dompierre 10
11. 11. Addition Rules Outline Multiplication RulesPrincipe of Inclusion-Exclusion for Four Sets n(A1 ∪ A2 ∪ A3 ∪ A4 ) = n(A1 ) + n(A2 ) + n(A3 ) + n(A4 ) − n(A1 ∩ A2 ) − n(A1 ∩ A3 ) − n(A1 ∩ A4 ) − n(A2 ∩ A3 ) − n(A2 ∩ A4 ) − n(A3 ∩ A4 ) + n(A1 ∩ A2 ∩ A3 ) + n(A1 ∩ A2 ∩ A4 ) + n(A1 ∩ A3 ∩ A4 ) + n(A2 ∩ A3 ∩ A4 ) − n(A1 ∩ A2 ∩ A3 ∩ A4 ) Julien Dompierre 11
12. 12. Addition Rules Outline Multiplication RulesPrincipe of Inclusion-Exclusion for n Sets Let A1 , A2 , ..., An be n ﬁnite sets. Then n(A1 ∪ A2 ∪ · · · ∪ An ) = n(Ai ) 1≤i≤n − n(Ai ∩ Aj ) 1≤i<j≤n + n(Ai ∩ Aj ∩ Ak ) 1≤i<j<k≤n − ··· + ··· − ··· n+1 + (−1) n(A1 ∩ A2 ∩ · · · ∩ An ) Julien Dompierre 12
13. 13. Addition Rules Outline Multiplication RulesOutline 1 Addition and Multiplication Rules for Probability Addition Rules Multiplication Rules Julien Dompierre 13
14. 14. Addition Rules Outline Multiplication RulesIndependent Events (p. 205) The multiplication rules can be used to ﬁnd the probability of two or more events that occur in sequence. For example, if a coin is tossed and then a die is rolled, one can ﬁnd the probability of getting a head on the coin and a 4 on the die. These two events are said to be independent since the outcome of the ﬁrst event (tossing a coin) does not aﬀect the probability outcome of the second event (rolling a die). Two events A and B are independent events if the fact that A occurs does not aﬀect the probability of B occurring. Julien Dompierre 14
15. 15. Addition Rules Outline Multiplication RulesMultiplication Rule 1 (p. 206) When two events are independent, the probability of both occurring is P(A ∩ B) = P(A) · P(B) Julien Dompierre 15
16. 16. Addition Rules Outline Multiplication RulesRemarks on the Multiplication Rule 1 1. Multiplication rule 1 can be extended to three or more independent events by using the formula P(A ∩ B ∩ C ∩ · · · ∩ K ) = P(A) · P(B) · P(C ) · · · P(K ) 2. In this sequence, the experiments may or may not be the same. If the experiments are the same, the events may or may not be the same. Julien Dompierre 16
17. 17. Addition Rules Outline Multiplication RulesDependent Events (p. 208) When the outcome or occurrence of the ﬁrst event A aﬀects the outcome or occurrence of the second event B in such a way that the probability is changed, the events A and B are said to be dependent events. The conditional probability of an event B in relationship to an event A is the probability that event B occurs given that the event A has already occurred. The notation for conditional probability is P(B|A). This notation does not mean that B is divided by A; rather, it means the probability that event B occurs given that event A has already occurred. Julien Dompierre 17
18. 18. Addition Rules Outline Multiplication RulesMultiplication Rule 2 (p. 208) When two events are dependant, the probability of both occurring is P(A ∩ B) = P(A) · P(B|A) Julien Dompierre 18
19. 19. Addition Rules Outline Multiplication RulesFormula for Conditional Probability (p. 210) The probability that the second event B occurs given that the ﬁrst event A has occurred can be found by dividing the probability that both events occurred by the probability that the ﬁrst event has occurred. The formula is P(A ∩ B) P(B|A) = P(A) Julien Dompierre 19
20. 20. Addition Rules Outline Multiplication RulesConditional Probability and Independent Events Two events A and B are independent if P(B|A) = P(B) and are dependent otherwise. Julien Dompierre 20
21. 21. Addition Rules Outline Multiplication RulesProbabilities for “At Least” or “At Most” (P. 213) In some case, it is easier to compute the probability of the complement of an event than the probability of the event itself. This is still true for a sequence of events. Example: A coin is tossed 5 times. Find the probability of getting at least one tail. This is equal to 1 minus the probability of getting no tail at all, which is all heads. Find the probability of getting at most four tails. This is equal to 1 minus the probability of getting ﬁve tails. Julien Dompierre 21