discrete-structures

# Theorems 1 let e be an event the probability of ē

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Theorems 1. Let E be an event. The probability of Ē , the complement of E , satisfies P(E) +P(Ē) = 1 . 2. Let E 1 and E 2 be events. Then P(E 1 U E 2) = P(E 1 ) + P(E 2 ) – P(E 1 ∩ E 2 ) note: If E 1 and E 2 are mutually exclusive events, P(E 1 U E 2) = P(E 1 ) + P(E 2 )

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PROBABILITY Exercises 1. Find errors in each of the ff assignments of probabilities. a. The probabilities that a student will have 0,1,2, or 3 or more mistakes are 0.51, 0.47, 0.24, and -0.12 respectively. b. The probability that it will rain tomorrow is 0.46 and the probability that it will not rain tomorrow is 0.55.
PROBABILITY 2. P(King or Spade) 3. P(King or Queen) 4. If two dice are thrown, what is the probability of obtaining sum of 7? of 5? 5. What is the probability of obtaining a sum of 7 or 5 if the two dice are thrown?

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PROBABILITY Definition(Conditional Probability) Let E and F be events with P(F)>0 . The conditional probability of E given F , denoted by P(E|F) is defined as P(E|F) = P(E∩F)/P(F) note: If E and F are independent events P(E) = P(E|F) The events E and F are independent if and only if: P(E∩F) = P(E)P(F)
PROBABILITY From a deck of 52 cards, P(Ace|red) = ? A Deck of 52 cards Type Color Total Red Black Ace 2 2 4 Non- Ace 24 24 48 Total 26 26 52

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PROBABILITY Example: What is the probability of getting a sum of 10, given that at least one die shows 5, when two fair dice are rolled?
PROBABILITY Solution: Let E be the event of getting a sum of 10 F be the event that at least one die shows 5 P(E∩F) – probability of getting a sum of 10 and at least one die shows 5 ( 29 ( 29 ( 29 11 1 36 11 36 1 = = = F P F E P F | E P

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PROBABILITY Example: Weather records shows that the probability of high barometric pressure is 0.80, and the probability of rain and high barometric pressure is 0.10. What is the probability of rain given high barometric pressure?
PROBABILITY Solution: R – denotes the event of “rain” H – denotes the event of “ high barometric pressure” ( 29 ( 29 ( 29 8 1 80 0 10 0 = = = . . H P H R P H | R P

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PROBABILITY Example (Independent Events) 1. Tossing a coin: What is the probability that head on first toss and tail on second? 2. Are the events that a family with 3 children has children of both sexes, and that a family with three children has at most one boy, independent? Assume that the eight ways a family can have 3 children are equally likely.
PROBABILITY 3. Joe and Olivia take a final exam in discrete structures. The probability that Joe passes is 0.70 and the probability that Olivia passes is 0.95. Assuming that the events “Joe passes” and “Olivia passes” are independent, find the probability that Joe or Olivia passes or both passes the final exam?

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PROBABILITY Baye’s Theorem Suppose that the possible classes are C 1 , …,C n . Suppose further that each pair of classes is mutually exclusive and each item to be classified belongs to one of the classes. For a feature set F , we have ( 29 ( 29 ( 29 ( 29 ( 29 = = n i i i j j j C P C | F P C P C | F P F | C P 1
PROBABILITY Example 1.

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