Module 5.3

• Discrete Probability Distributions


  • Binomial distribution
  • Geometric distribution
  • Poisson distribution
  • Hypergeometric distribution
  • Negative binomial distribution
• Binomial Experiments


  • The experiment is repeated for a fixed number of trials, where each trial is independent of other trials
  • There are only two possible outcomes of interest for each trial, classified as a success (S) or as a failure (F)
  • The probability of a success P (S) is the same for each trial
  • The random variable x counts the number of successful trials
• n


  The number of times a trial is repeated
• p = P (S)


  The probability of success in a single trial
• q = P (F)


  The probability of failure in a single trial (q = 1 - p)
• x


  The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3, ..., n
• Binomial Experiment


  • You randomly select a card from a deck of cards, and note if the card is an Ace. You then put the card back and repeat this process 8 times.
• This is a binomial experiment. Each of the 8 selections represent an independent trial because the card is replaced before the next one is drawn. There are only two possible outcomes: either the card is an Ace or not.
• Binomial Experiment


  • You roll a die 10 times and note the number the die lands on.
• This is not a binomial experiment. While each trial (roll) is independent, there are more than two possible outcomes: 1, 2, 3, 4, 5, and 6.
• Binomial Experiment


  • A bag contains 10 chips. 3 of the chips are red, 5 of the chips are white, and 2 of the chips are blue. Three chips are selected, with replacement. Find the probability that you select exactly one red chip.
• The binomial probability distribution is: P(0) = 0.240, P(1) = 0.412, P(2) = 0.265, P(3) = 0.076, P(4) = 0.008
• P(no more than 3) = P(x <= 3) = 0.993
• P(at least 1) = P(x >= 1) = 1 - P(0) = 0.76
• The graph of the binomial probability distribution is a histogram.
• Mean (μ)


  np
• Variance (σ^2)


  npq
• Standard Deviation (σ)


  sqrt(npq)
• Binomial Experiment


  • One out of 5 students at a local college say that they skip breakfast in the morning. Find the mean, variance and standard deviation if 10 students are randomly selected.
• Standard Deviation (σ) = sqrt(1.6) = 1.3
• Poisson Distribution


  • The experiment consists of counting the number of times an event, x, occurs in a given interval
  • The probability of the event occurring is the same for each interval
  • The number of occurrences in one interval is independent of the number of occurrences in other intervals
• Poisson Probability Formula


  P(x) = (e^(-μ) * μ^x) / x!
• Poisson Distribution


  • The mean number of power outages in the city of Brunswick is 4 per year. Find the probability that in a given year, there are exactly 3 outages, and the probability that there are more than 3 outages.