Business Statistics: A Decision-Making Approach 6th EditionChapter 5 Discrete and Continuous Probability DistributionsChap 5-1Chapter GoalsAfter completing this chapter, you should be able to: nApply the binomial distribution to applied problemsnCompute probabilities for the Poisson and hypergeometric distributionsnFind probabilities using a normal distribution table and apply the normal distribution to business problemsnRecognize when to apply the uniform and exponential distributions2Probability DistributionsContinuous Probability DistributionsBinomialHypergeometricPoissonProbability DistributionsDiscrete Probability DistributionsNormalUniformExponential3nA discrete random variable is a variable that can assume only a countable number of valuesMany possible outcomes:n number of complaints per dayn number of TVs in a householdn number of rings before the phone is answeredOnly two possible outcomes:n gender: male or femalen defective: yes or non spreads peanut butter first vs. spreads jelly firstDiscrete Probability Distributions4Continuous Probability DistributionsnA continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values)nthickness of an itemntime required to complete a taskntemperature of a solutionnheight, in inchesnThese can potentially take on any value, depending only on the ability to measure accurately.5The Binomial DistributionBinomialHypergeometricPoissonProbability DistributionsDiscrete Probability Distributions6The Binomial DistributionnCharacteristics of the Binomial Distribution:nA trial has only two possible outcomes “success” or “failure”nThere is a fixed number, n, of identical trialsnThe trials of the experiment are independent of each othernThe probability of a success, p, remains constant from trial to trialnIf p represents the probability of a success, then (1-p) = q is the probability of a failure7Binomial Distribution SettingsnA manufacturing plant labels items as either defective or acceptablenA firm bidding for a contract will either get the contract or notnA marketing research firm receives survey responses of “yes I will buy” or “no I will not”nNew job applicants either accept the offer or reject it8Counting Rule for CombinationsnA combination is an outcome of an experiment where x objects are selected from a group of n objectswhere: n! =n(n - 1)(n - 2) . . . (2)(1)x! = x(x - 1)(x - 2) . . . (2)(1)0! = 1 (by definition)9P(x) = probability of x successes in n trials,with probability of success p on each trialx = number of successes in sample, (x = 0, 1, 2, ., n)p = probability of “success” per trialq = probability of “failure” = (1 p)n = number of trials (sample size)P(x)n x ! nxp qxnx! ()!=-Example: Flip a coin four times, let x = # heads:n = 4p = 0.5q = (1 - .5) = .5x = 0, 1, 2, 3, 4Binomial Distribution Formula10n = 5 p = 0.1n = 5 p = 0.5Mean0.2.4.6012345XP(X).2.4.6012345XP(X)0Binomial DistributionnThe shape of the binomial distribution depends on the values of p and n Here, n = 5 and p = .1 Here, n = 5 and p = .511Binomial Distribution CharacteristicsnMeannVariance and Standard DeviationWheren = sample size p = probability of success q = (1 p) = probability of failure 12n = 5 p = 0.1n = 5 p = 0.5Mean0.2.4.6012345XP(X).2.4.6012345XP(X)0Binomial CharacteristicsExamples13Using Binomial Tablesn = 10 xp=.15p=.20p=.25p=.30p=.35p=.40p=.45p=.50 0 1 2 3 4 5 6 7 8 9 100.1969 0.3474 0.2759 0.1298 0.0401 0.0085 0.0012 0.0001 0.0000 0.0000 0.00000.1074 0.2684 0.3020 0.2013 0.0881 0.0264 0.0055 0.0008 0.0001 0.0000 0.00000.0563 0.1877 0.2816 0.2503 0.1460 0.0584 0.0162 0.0031 0.0004 0.0000 0.00000.0282 0.1211 0.2335 0.2668 0.2001 0.1029 0.0368 0.0090 0.0014 0.0001 0.00000.0135 0.0725 0.1757 0.2522 0.2377 0.1536 0.0689 0.0212 0.0043 0.0005 0.00000.0060 0.0403 0.1209 0.2150 0.2508 0.2007 0.1115 0.0425 0.0106 0.0016 0.00010.0025 0.0207 0.0763 0.1665 0.2384 0.2340 0.1596 0.0746 0.0229 0.0042 0.00030.0010 0.0098 0.0439 0.1172 0.2051 0.2461 0.2051 0.1172 0.0439 0.0098 0.001010 9 8 7 6 5 4 3 2 1 0 p=.85p=.80p=.75p=.70p=.65p=.60p=.55p=.50xExamples: n = 10, p = .35, x = 3: P(x = 3|n =10, p = .35) = .2522n = 10, p = .75, x = 2: P(x = 2|n =10, p = .75) = .000414Using PHStatnSelect PHStat / Probability & Prob. Distributions / Binomial15Using PHStatnEnter desired values in dialog boxHere: n = 10 p = .35Output for x = 0 to x = 10 will be generated by PHStatOptional check boxes for additional output16P(x = 3 | n = 10, p = .35) = .2522PHStat OutputP(x 5 | n = 10, p = .35) = .094917The Poisson DistributionBinomialHypergeometricPoissonProbability DistributionsDiscrete Probability Distributions18The Poisson DistributionnCharacteristics of the Poisson Distribution:nThe outcomes of interest are rare relative to the possible outcomesnThe average number of outcomes of interest per time or space interval is nThe number of outcomes