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Section 5.2

Section 5.2. Some Probability Rules Compound Events. Independent Events. The occurrence (or non-occurrence) of one event does not change the probability that the other event will occur. If events A and B are independent , P ( A and B ) = P ( A ) • P ( B )

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Section 5.2

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  1. Section 5.2 Some Probability Rules Compound Events

  2. Independent Events • The occurrence (or non-occurrence) of one event does not change the probability that the other event will occur. • If events A and B are independent, • P(A and B) = P(A) •P(B) • Example: Rolling a dice two times. The outcome of the first roll does not have any affect on the probability of the second roll. Because of this the two events are independent.

  3. Conditional Probability • If events are dependent, the fact that one occurs affects the probability of the other. • P(A, givenB) equals the probability that event A occurs, assuming that B has already occurred.

  4. General Multiplication Rule for Any Events: P(A and B) = P(A) •P(B, given that A has occurred) P(Aand B) = P(B) •P(A, given that B has occurred) The Multiplication Rules: For independent events: P(A and B) = P(A) •P(B) For any events: P(A and B) = P(A) •P(B, given A) P(A and B) = P(B) •P(A, given B)

  5. The Multiplication Rules: • For independent events: P(A and B) = P(A) •P(B) • For any events: P(A and B) = P(A) •P(B, given A) P(A and B) = P(B) •P(A, given B)

  6. For independent events:P(A and B) = P(A) •P(B) When choosing two cards from two separate decks of cards, find the probability of getting two fives. P(two fives) = P(5 from first deck and 5 from second) =

  7. For dependent events:P(A and B) = P(A) · P(B, given A) When choosing two cards from a deck without replacement, find the probability of getting two fives. P(two fives) = P(5 on first draw and 5 on second) = P(5) P(5, given 5 occurred) =

  8. Multiplication Rules The multiplication rules apply whenever we wish to determine the probability of two events happening together. To indicate together, we use “and” between the events. But before we use the multiplication rule to compute the probability of A and B, you must determine if A and B are independent or dependent events.

  9. “And” versus “or” And means both events occur together. • The Event A and B Ormeans that at least one of the events occur. The Event A or B

  10. General Addition Rule For any eventsA and B, P(A or B) = P(A) + P(B) – P(A and B)

  11. When choosing a card from an ordinary deck, the probability of getting a five or a red card: P(5 or red) = P(5 ) + P(red) – P(5 and red) =

  12. When choosing a card from an ordinary deck, the probability of getting a five or a six: P(5 or 6) = P(5 ) + P(6) – P(5 and 6) =

  13. Mutually Exclusive Events • Events that cannot happen together. P(A and B) = 0 For any mutually exclusive events A and B, P(A or B) = P(A) + P(B) Example: When rolling an ordinary die: P(4 or 6) = P(4) + P(6) =

  14. How to use the addition rules 1/ Determine whether A and B are mutually exclusive events. If p(A and B) = 0, then the events are mutually exclusive. 2/ If A and B are mutually exclusive, P(A or B) = P(A) + P(B) 3/ If A and B are any events, P(A or B) = P(A) + P(B) – P(A and B)

  15. ExampleMutually exclusive Events Laura is playing monopoly. On her next move she needs to throw a sum bigger than 8 on the two dice in order to land on her own property and pass Go. What is the probability that Laura will roll a sum bigger than 8? Solution: The probability of throwing more than 8 is the same as P(9 or 10 or 11 or 12)

  16. Example Cont. Since the events are mutually exclusive, P(9 or 10 or 11 or 12) = P(9)+P(10)+P(11)+P(12)= For the sample space there are 36 equally likely outcomes. For example the those favorable to 9 are: 6 , 3; 3 , 6; 5 , 4; and 4 , 5. So, P(9) = 4/36. The other values can be computed in a similar way.

  17. Survey results:

  18. Survey results cont. Find: a/ P(male and college grad) b/ P(male or college grad) c/ P(male, given college grad) Solution: a/ P(male and college grad) = b/ P(male or college grad) = P(male) + P(Col. Grad.) - P(male and col. Grad) = c/ P(male, given college grad) =

  19. Summaryof basic probability rules A statistical experiment or statistical observation is any random activity that results in a recordable outcome. The sample space is the set of all simple events that are the outcomes of the statistical experiment and cannot be broken into other “simpler” events. A general even is any subset of the sample space. The notation P(A) designates the probability of the event A. • P(entire sample space) = 1 • For any event A: • designates the complement of A:

  20. Summary cont. • Events A and B are independent events if P(A) = P(A, given B) • Multiplication rules General: P(A and B) = P(A) * P(B, given A) Independent events: P(A and B) = P(A) * P(B) • Conditional Probability: • Events A and B are mutually exclusive if P(A and B) = 0 • Addition Rules. General: P(A or B) = P(A) + P(B) – P(A and B) Mutually exclusive events: P(A or B) = P(A) + P(B) Assignments 10, 11 and 12

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