CHAPTER 5 Discrete Probability Distribution Student Notes Stat 200 Elementary Statistics Introduction This chapter explains the concepts and applications of probability distributions In addition special probability distributions such as the binomial multinomial Poisson and hypergeometric distributions are explained Section 5 1 Probability Distribution Random variable Example Construct a probability distribution for flipping three coins Discrete Probability Distribution Example Construct a probability distribution for rolling a single die Graphing the probability N of heads 0 1 2 3 Probability Two Requirements for a Probability Distribution 1 2 Example Write the distribution for the formula P X 1 X where X 2 3 6 Section 5 2 Mean Variance Standard Deviation and Expectation Calculating Mean Examples 1 Find the mean of the number of spots that appear when a die is tossed 2 In a family with two children find the mean of the number of children who will be girls 3 If three coins are tossed find the mean of the number of heads that occur Variance of a Probability Distribution s 2 X 2 P X m2 2 Examples Compute the variance and standard deviation of the number of spots that appear when a die is tossed Five balls numbers 0 2 4 6 and 8 are placed in a bag After the balls are mixed one is selected its number is noted and then it is replaced If this experiment is repeated many times find the variance and standard deviation of the numbers on the balls Expected Value Example One thousand tickets are sold at 1 each for a color television valued at 350 What is the expected value of the gain if a person purchases one ticket Section 5 3 Binomial Distribution The binomial experiment is a probability experiment that satisfies these requirements 1 Each trial can have only two possible outcomes or 2 There must be a number of trials 3 The outcomes of each trial must be of each other 4 The probability of success must remain the for each trial Notation for the Binomial Distribution Binomial Probability Formula P X n p X q n X n X X Examples A coin is tossed three times Find the probability of getting exactly two heads A survey found that one out of five Americans say he or she has visited a doctor in any given month If 10 people are selected at random find the probability that exactly 3 will have visited a doctor last month A coin is tossed three times Find the probability of getting exactly two heads using table B p 756 Binomial Distribution Properties Mean Variance Standard Deviation A coin is tossed 4 times Find the mean variance and standard deviation of the number of heads that will be obtained A die is rolled 480 times Find the mean variance and standard deviation of the number of 2s that will be rolled
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