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Understanding Discrete Probability Distributions and Binomial Experiments

This chapter focuses on discrete probability distributions, emphasizing random variables and their outcomes in probability experiments. It differentiates between discrete and continuous random variables, providing examples such as the number of books in a library and snowfall amounts. Key concepts include the formation of probability distributions, mean, variance, and standard deviation calculations. It also introduces binomial distributions characterized by fixed trials and consistent success probabilities, along with examples for applying the binomial probability formula in real-world scenarios.

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Understanding Discrete Probability Distributions and Binomial Experiments

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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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