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Probability Part I

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Probability Probability refers to the chances of an event happening. Symbolize P(A) to refer to event A.

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Values of Probability All values are between 0 and 1. Write answers as 3 place decimals. If P(A) = 0, it means the event WILL NOT happen. If P(A) = 1, it means the event WILL happen.

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Relative frequency approximation of probability Conduct an experiment a large number of times.

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Example A study of 150 randomly selected American Airline flights showed that 108 arrived on time. What is the probability that a randomly selected flight will arrive late?

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Solution Number of late flights is 150 – 108 or 42. P(late) = 42/150 P(late) = 0.280

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Theoretical probability Based upon what should occur, if events are equally likely.

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Example A die is rolled. What is the probability that the number showing is greater than 2?

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Solution There are 6 sides on a die. Four numbers (3,4,5 and 6) are greater than 2. P(A) = 4/6 P(A) =.667

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Example A card is selected from a standard deck of cards. What is the probability that it is a King?

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Solution A deck of cards has 52 cards. There are 4 Kings. P(King) = 4/52 P(King) = 0.077

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Law of Large numbers If an experiment is repeated again and again, the relative frequency probability of an event tends to approach the actual probability.

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Complement of an event Symbolized: Represents the chances an event will not happen Found by:

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Example For the airline example, we could say: P(on time) = 108/150 P(on time) = 0.720 P(late) = 1 – P(on time) P(late) = 1 – 0.720 P(late) = 0.280

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Odds Against an event In favor of an event

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Example The probability that a horse will win a race is 2/7. Find the odds against the horse winning the race.

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Solution P(lose) = 1 – 2/7 P(lose) = 5/7 Odds against = Odds against = 5:2

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Payoff odds Payoff odds against event A = amount of net profit : amount bet

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Example The odds against a horse winning is 5:2. You bet $15 on the horse. What is your net profit, if the horse wins?

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Solution 5 : 2 = net profit : 15 Write: 2x = 75 X = $37.50 Get back $52.50 ($37.50 + $15)

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Addition Rule for 2 events P(A or B) means P(event A happens OR event B happens OR they both happen) P(A or B) = P(A) + P(B) – P(A and B)

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Example A card is drawn from a deck of 52 cards. What is the probability that it is either a King or a Heart?

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Solution There are 4 Kings. There are 13 Hearts. There is 1 King of Hearts.

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Solution There are 4 Kings. There are 13 Hearts. There is 1 King of Hearts. P(King or Heart) = 4/52 + 13/52 – 1/52 P(King or Heart) = 16/52 P(King or Heart) = 0.308

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Example A card is drawn from a deck of 52 cards. What is the probability it is a King or a Queen?

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Solution There are 4 Kings. There are 4 Queens. A card can not be a King and a Queen.

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Solution There are 4 Kings. There are 4 Queens. A card can not be a King and a Queen. P(King or Queen) = 4/52 + 4/52 – 0/52 P(King or Queen) = 8/52 P(King or Queen) = 0.154

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Mutually Exclusive Two events are mutually exclusive if they can not both occur at the same time. P(A and B) = 0 In the example, getting a King AND a Queen are mutually exclusive.

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Tables to present data A sample of 1000 people was obtained. There were 500 men and 500 women. Of the men, 63 were left handed. Of the women, 50 were left handed. MenWomenTotal Left handed6350113 Right handed437450887 Total500 1000

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Questions: If a person is randomly selected, what is the probability that: a. The person is a man b. The person is left handed c. The person is a left handed man d. The person is a man or is left handed

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Solution The person is a man P(man) = 500/1000 P(man) = 0.500 The person is left handed P(left handed) = 113/1000 P(left handed) = 0.113

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Solution The person is a left handed man P(left handed and man) = 63/1000 P(left handed and man) = 0.063 The person is a man or is left handed P(man or left handed) =.500 +.113 -.063 P(man or left handed) = 0.550

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