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Probability

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

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Probability

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  1. Probability Part I

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

  3. 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.

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

  5. 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?

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

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

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

  9. 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

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

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

  12. 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.

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

  14. 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

  15. Odds • Against an event • In favor of an event

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

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

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

  19. 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?

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

  21. 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)

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

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

  24. 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

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

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

  27. 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

  28. 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.

  29. 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.

  30. 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

  31. 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

  32. 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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