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MDM4U1 5.1 Probability Distribution Probability Distribution: In this chapter we will develop models for distributions that show the probabilities of all possible outcomes in an experiment. For example, in an experiment where a die is rolled, the probability distribution would include the probability of each possible outcome (1, 2, 3, 4, 5, and 6). In this example “the number rolled with the die” is the random variable. Throughout this unit we will be looking at 4 types of probability distributions: 1) 2) 3) 4) Uniform Distribution Binomial Distribution Geometric Distribution Hypergeometric Distribution This unit describes probability distributions involving discrete random variables. Recall: Discrete Variables: Continuous Variables: Example 1 Classify the following random variables as discrete or continuous. a) Length of time you stay in class b) Number of courses you take in a semester Uniform distribution is the distribution of probabilities with equally likely outcomes. Uniform random variable with ‘n’ outcomes 𝑃(𝑥) = 1 𝑛 Example 2 Determine whether the following experiments have a uniform probability a) Guessing a person’s age b) Rolling a number on a die Example 3 Uniform Probability Distribution Determine the probability distribution for rolling a number on a dice. x 1 2 3 4 5 6 P(x) Expected Value is the average of the outcomes. “Expected value” or “Expectation” 𝐸(𝑥) = 𝑥1 𝑃(𝑥1 ) + 𝑥2 𝑃(𝑥2 ) + ⋯ + 𝑥𝑛 𝑃(𝑥𝑛 ) 𝑛 𝐸(𝑥) = ∑ 𝑥𝑖 𝑃(𝑥𝑖 ) 𝑖=1 Example 4 Given the following probability distribution, determine the expected value. x 2 4 6 P(x) 0.4 0.1 0.5 Example 5 a) Determine the probability distribution for the sum rolled with two dice. 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 b) Determine the expected sum of two dice. x 1 2 3 4 5 6 7 8 9 10 11 12 P(x)