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Unit 3: Probability You will need to be able to describe how you will perform a simulation Create a correspondence between random numbers and outcomes Explain how you will generate your random numbers and when you will know to stop Make sure you understand the purpose of the simulation With or without replacement? Probability is a measurement of the likelihood of an event. It represents the proportion of times we’d expect to see an outcome in a long series of repetitions. P(event) # 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠 = # 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 Addition Rule: 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵) Multiplication Rule: 𝑃 𝐴 ∩ 𝐵 = 𝑃(𝐴) × 𝑃(𝐵|𝐴) Conditional Probability: 𝑃 𝐴𝐵 = 𝑃(𝐴∩𝐵) 𝑃(𝐵) Disjoint (mutually exclusive) basically means no outcomes in common. Two events are independent if the occurrence of one has no effect on the probability that the other event occurs. Two tests for independence: 𝑃 𝐴 𝐵 = 𝑃 𝐴 or 𝑃 𝐴 𝑎𝑛𝑑𝐵 = 𝑃(𝐴) × 𝑃(𝐵) When calculating probabilities, it helps to consider the Sample Space. List all outcomes if possible. Draw a tree diagram or Venn diagram Use the Multiplication Counting Principle Sometimes it is easier to use common sense rather than memorizing formulas! This chapter introduced us to the concept of a random variable. We learned how to describe an expected value and variability of both discrete and continuous random variables. A Random Variable, X, is a variable whose outcome is unpredictable in the short-term, but shows a predictable pattern in the long run. Discrete vs. Continuous The Expected Value, E(X)=μ, is the long-term average value of a Random Variable. E(X) for a Discrete X E(X) = μ = 𝑥 ∙ 𝑝(𝑥) X 1 5 20 P(X) 0.5 0.2 0.3 μ = 1(0.5) + 5(0.2) + 20(0.3) = .5 + 1+ 6 = 7.5 The Variance, 𝑉𝑎𝑟(𝑋) = 𝜎 2 , is the amount of variability from μ that we expect to see in X. The Standard Deviation of X, 𝜎 = X 1 5 20 P(X) 0.5 0.2 0.3 𝑉𝑎𝑟 𝑋 = 𝜎 2 = 𝑥−𝜇 𝑉𝑎𝑟 𝑋 = 69.25 𝜎 = 8.32 2 ∙ 𝑝(𝑥) 𝑉𝑎𝑟(𝑋) The following rules are helpful when working with Random Variables. 𝜇𝑎+𝑏𝑋 = 𝑎 + 𝑏𝜇𝑥 𝜇𝑋±𝑌 = 𝜇𝑋 ± 𝜇𝑌 2 𝜎𝑎+𝑏𝑋 = 𝑏2 𝜎𝑥2 2 𝜎𝑋±𝑌 = 𝜎𝑋2 + 𝜎𝑌2 Some Random Variables are the result of events that have only two outcomes (success and failure). We define a Binomial Setting to have the following features Two Outcomes - success/failure Fixed number of trials - n Independent trials Equal P(success) for each trial If X is B(n,p), the following formulas can be used to calculate the probabilities of events in X. P(X = k) = binompdf (n, p, k) P(X ≤ k) = binomcdf (n, p, k) If conditions are met, a binomial situation may be approximated by a normal distribution If np>10 and n(1-p)>10, then B(n,p) ~ Normal Some Random Variables are the result of events that have only two outcomes (success and failure), but have no fixed number of trials. We define a Geometric Setting to have the following features: Two Outcomes - success/failure No Fixed number of trials Independent trials Equal P(success) for each trial If X is Geometric, the following formulas can be used to calculate the probabilities of events in X. 𝑃 𝑋 =𝑘 = 1−𝑝 1 𝜇𝑥 = 𝑝 𝜎𝑥 = 𝑘−1 𝑝 = 𝑔𝑒𝑜𝑚𝑒𝑡𝑝𝑑𝑓(𝑝, 𝑘) 1−𝑝 𝑝2 𝑃 𝑋 >𝑘 = 1−𝑝 𝑘 A sampling distribution is the distribution of all samples of size n taken from the population. Be able to describe center, shape and spread When we take a sample, we are not guaranteed the statistic we measure is equal to the parameter in question. Further, repeated sampling may result in different statistic values. Bias and Variability The center of the sampling distribution is equal to the population mean The shape of the sampling distribution is approximately Normal: If the population is approximately Normal OR The sampling distribution becomes more normal as the sample size increases (CLT) The standard deviation of the sampling distribution is called the Standard Error (SE) and 𝜎 can be found by 𝜎𝑋 = 𝑛 Can use Normal techniques to solve problems 𝑥−𝑚𝑒𝑎𝑛 𝑥−𝜇 with 𝑧 = = 𝜎 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑛 The center is equal to p, the population proportion. The standard deviation is 𝜎𝑝 = 𝑝(1−𝑝) 𝑛 As the sample size gets larger, the shape of the sampling distribution gets more normal. np > 10 and n(1-p) > 10 Can use normal distribution as an 𝑝−𝑝 approximation with 𝑧 = 𝑝(1−𝑝) 𝑛