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4.5 – Comparing Probability Distributions In earlier chapters, you looked at independent and dependent events. When determining the appropriate probability distribution, this is an important criterion to consider. How would you differentiate between independent and dependent events? The following chart summarizes the general conditions of the distributions: Parameters and what they represent Definition of random variable, x Range of values for x Probability formula Expectation formula Identifying characteristics Uniform 𝑛 = number of items Value of the outcome Depends on the situation 1 𝑃(𝑥) = 𝑛 𝐸(𝑋) = 1 ∑𝑥 𝑛 All items are equally likely A single trial Binomial 𝑛 = number of trials 𝑝 = probability of success on an individual trial 𝑞 = probability of failure on an individual trial Number of successful Outcomes 𝑥 = 0, 1, 2, … , 𝑛 𝑛 𝑃(𝑥) = ( ) (𝑝)𝑘 (1 − 𝑝)𝑛−𝑘 𝑘 Hypergeometric 𝑛 = size of the population 𝑟 = number of trials 𝑎 = number of successful items available 𝐸(𝑋) = 𝑛𝑝 𝐸(𝑥) = Trials are independent Successes are counted Number of successful Outcomes 𝑥 = 0, 1, 2, … , 𝑟 (𝑎)(𝑛−𝑎) 𝑃(𝑥) = 𝑥 𝑛𝑟−𝑥 (𝑟 ) 𝑟𝑎 𝑛 Trials are dependent Successes are counted Compare and contrast the general conditions for the binomial and hypergeometric probability distributions: Compare and Contrast Concept 1 Binomial Distribution Concept 2 Hypergeometric Distribution How are they alike? Random variable, x, is the number of successes. Fixed number of trials. Successes are counted. Distributions is discrete How are they different? Number of trials Independent trials Size of population Dependent trials 𝑥 = 0, 1, 2, … , 𝑛 Probability of success does not change 𝐸(𝑋) = 𝑛𝑝 𝑥 = 0, 1, 2, … , 𝑟 Probability of success changes in every trial 𝑟𝑎 𝐸(𝑥) = 𝑛 Parameters: 𝑛, 𝑝, 𝑞 Parameters: 𝑛, 𝑟, 𝑎 Mathematics of Data Management (MDM4UC) Page 1 4.5 – Comparing Probability Distributions Use the following flow chart to help you decide when to use each type of distribution (uniform, binomial, hypergeometric, neither). Practice: (Page 185) #1-4, 6 Mathematics of Data Management (MDM4UC) Page 2