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Probability II

Probability II. Denoted by P(Event). Probability. This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely. The relative frequency at which a chance experiment occurs Flip a fair coin 30 times & get 17 heads.

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Probability II

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  1. Probability II

  2. Denoted by P(Event) Probability This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely.

  3. The relative frequency at which a chance experiment occurs Flip a fair coin 30 times & get 17 heads Experimental Probability

  4. Rule 1.Legitimate Values For any event E, 0 < P(E) < 1 Rule 2.Sample space If S is the sample space, P(S) = 1 Basic Rules of Probability

  5. Rule 3.Complement • For any event E, • P(E) + P(not E) = 1

  6. Rule 4.Addition • If two events E & F are disjoint, • P(E U F) = P(E) + P(F) • (General) If two events E & F are not disjoint, • P(E U F) = P(E) + P(F) – P(E ∩ F)

  7. Ex 1) A large auto center sells cars made by many different manufacturers. Three of these are Honda, Nissan, and Toyota. Suppose that P(H) = .25, P(N) = .18, P(T) = .14. Are these disjoint events? yes P(H U N U T) = .25 + .18+ .14 = .57 P(not (H U N U T) = 1 - .57 = .43

  8. Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs The probability that a coin toss results in tails, given it is a cloudy day? The probability that the sidewalk is wet, given it is a cloudy day? Independent Independent Not independent

  9. Rule 5.Multiplication If two events A & B are independent, P(A∩B) = P(A) • P(B)

  10. Ex. 3) A certain brand of firecracker claims that it will go off 95% of the time. You buy three firecrackers, what is the probability that all go off? (assume they are independent)

  11. Ex. 3) A certain brand of firecracker claims that it will go off 95% of the time. You buy three firecrackers, what is the probability that all go off? (assume they are independent) P(S∩S ∩S) =

  12. Ex. 3) A certain brand of firecracker claims that it will go off 95% of the time. You buy three firecrackers, what is the probability that all go off? (assume they are independent) P(S∩S) = (0.95)(0.95)(0.95) = 0.8574

  13. Rule 6.At least one The probability that at least one outcome happens is 1 minus the probability that no outcomes happen. P(at least 1) = 1 – P(none)

  14. Dr. Pepper Ex 4) For a sales promotion the manufacturer places winning symbols under the caps of 10% of all Dr. Pepper bottles. You buy a six-pack. What is the probability that you win something? Let a win = success (S) Let no win = failure (F) If P(S) = .10, then P(F) = .90 P(at least one winning symbol) = 1 – P(no winning symbols) 1 - .96 = .4686

  15. A probability that takes into account a given condition Rule 7: Conditional Probability

  16. Ex 5) In a recent study it was found that the probability that a randomly selected student is a girl is .51 and is a girl and plays sports is .10. If the student is female, what is the probability that she plays sports?

  17. Ex 6) The probability that a randomly selected student plays sports if they are male is .31. What is the probability that the student is male and plays sports if the probability that they are male is .49?

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