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chAPTER four

chAPTER four. Discrete Probability Distributions. Section 4.1. Probability Distributions. Random Variables. A random variable x represents a numerical value associated with each outcome of a probability experiment. It is DISCRETE if it has a finite number of possible outcomes.

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chAPTER four

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  1. chAPTER four Discrete Probability Distributions

  2. Section 4.1 Probability Distributions

  3. Random Variables • A random variable x represents a numerical value associated with each outcome of a probability experiment. • It is DISCRETE if it has a finite number of possible outcomes. • It is CONTINUOUS if it has an uncountable number of possible outcomes (represented by an interval)

  4. EX: Discrete or Continuous? • 13. the number of books in a university library. • 19. the amount of snow (in inches) that fell in Nome, Alaska last winter.

  5. Discrete Probability Distributions • - list of each possible value and its probability. Must satisfy 2 conditions: • 1. 0 < P(x) < 1 • 2. ΣP(x) = 1

  6. EX: make a probability distribution • 28. the # of games played in the World Series from 1903 to 2009

  7. To Find… • MEAN µ = Σ[x·P(x)] • VARIANCE σ2 = Σ[(x - µ)2·P(x)] • STANDARD DEVIATION σ = √σ 2

  8. Find µ, σ2, and σ • 36. The # of 911 calls received per hour.

  9. Expected Value Notation: E(x) Expected value represents what you would expect to happen over thousands of trials. SAME as the MEAN!!!E(x) = µ = Σ[x·P(x)]

  10. EX: find expected NET GAIN • If x is the net gain to a player in a game of chance, then E(X) is usually negative. This value gives the average amount per game the player can expect to lose. • 46. A charity organization is selling $5 raffle tickets. First prize is a trip to Mexico valued at $3450, second prize is a spa package valued at $750. The remaining 20 prizes are $25 gas cards. The number of tickets sold is 6000.

  11. Section 4.2 Binomial Distributions

  12. Binomial Experiments • CONDITIONS: • 1. there are a fixed number of independent trials (n = # of trials) • 2. Two possible outcomes for each trial, Success or Failure. • 3. Probability of Success is the same for each trial. p = P(Success) and q = P(Failure) • 4. random variable x = # of successful trials

  13. Ex (from p 212) If binomial, ID ‘success’, find n, p, q; list possible values of x. If not binomial, explain why. • 10. From past records, a clothing store finds that 26% of people who enter the store will make a purchase. During a one-hour period, 18 people enter the store. The random variable represents the # of people who do NOT make a purchase.

  14. Binomial Probability Formula • To find the probability of (exactly) x number of successful trials: • P(x) = nCx · px · qn –x

  15. EX: Find the indicated probability • 18. A surgical technique is performed on 7 patients. You are told there is a 70% chance of success. Find the probability that the surgery is successful for • A) exactly 5 patients • B) at least 5 patients • C) less than 5 patients

  16. To Find… • MEAN µ = np • VARIANCE σ 2 = npq • STANDARD DEVIATION σ = √σ 2

  17. Construct a probability distribution, then find mean, variance, and standard deviation for the following: • 28. One in four adults claims to have no trouble sleeping at night. You randomly select 5 adults and ask them if they have trouble sleeping at night.

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