. . . PENNIES for the AGES . . .

1. . . . PENNIES for theAGES. . . • Push the “Sample More Data” button on the screen and read the average age of a sample of 64 pennies taken from the jar. • Note the horizontal and vertical scales on the grid here and then record that (rounded) average age using properly scaled X. MAT 312

2. Prob & Stat (MAT 312)Dr. Day Tuesday April 22, 2014 • Grab 64 Pennies at Random then Plot Average Age • Simulations Review • Probability Distributions: Sample Spaces & Random Variables • Revisit: Normal Distributions • Collect Assignment #9 MAT 312

3. Probability Simulations Simulations provide a means for calculating probabilities in situations where time, money, risk of injury, or other factors compel us to NOT carry out a real-life experiment of the situation. MAT 312

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6. Probability Distributions A probability distribution describes all possible outcomes of a probabilistic situation together with the probability associated with each outcome. When the set of outcomes is described numerically, we call those outcomes random variables. MAT 312

7. Probability Distributions Themeanorexpected valueof a probability distribution isthe long-run average value of the outcome of a probabilistic situation; if X is a random variable with values a(1), a(2), . . . , a(n) and associated probabilities p(1), p(2), p(3). . . , p(n), then the expected value of X is: E(X) = a(1)*p(1)+ a(2)*p(2) + . . . + a(n)*p(n). MAT 312

8. Probability Distributions Thevarianceof a probability distribution is a measure of the spread of the distribution; if the values of a random variable X are a(1), a(2), . . . , a(n), with associated probabilities p(1), p(2), p(3). . . , p(n), then the variance of the random variable X is: V(X) = var(X) = (a(1)-E(X))2*p(1) + (a(2)-E(X))2*p(2) + . . . + (a(n)-E(X))2*p(n). Note that the standard deviation of a distribution is just the square root of its variance. MAT 312

9. Common Probability Distributions Uniform Distributions: Probability distributions for which every outcome is equally likely. Outcomes for a single roll of one fair die form a uniform distribution. MAT 312

10. Common Probability Distributions Binomial Distributions: Probability distributions for which all four of the following properties are true: • There are exactly two outcomes to each trial, typically referred to as success and failure. • The total number of trials is fixed in advance. • The outcomes from trial to trial areindependent of each other. • The probability of success is the same from trial to trial. MAT 312

11. Binomial Distributions Examples • Flipping a fair coin 20 times and recording each outcome as heads or tails. • Rolling a die 10 times and recording each result as 6 or not 6. • Grabbing, with replacement, a block from a bag of blocks and recording that the block is green or not green. Repeat 100 times. MAT 312

12. Common Probability Distributions Normal Distributions: Continuous probability distributions that are symmetric and mound shaped, characterized with a mean (μ) and a standard deviation (σ). Many phenomena can be represented using a normal distribution. MAT 312

13. Terms, Symbols, & Properties • outcomes: the possible results of an experiment • equally likely outcomes: a set of outcomes that each have the same likelihood of occurring • sample space: the set of all possible outcomes to an experiment • uniform sample space: a sample space filled with equally likely outcomes • non-uniformsample space: a sample space that contains two or more outcomes that are not equally likely • event: a collection of one or more elements from a sample space MAT 312

14. expected value: the long-run average value of the outcome of a probabilistic situation; if an experiment has n outcomes with values a(1), a(2), . . . , a(n), with associated probabilities p(1), p(2), p(3). . . , p(n), then the expected value of the experiment is a(1)*p(1)+ a(2)*p(2) + . . . + a(n)*p(n). • random event: an experimental event that has no outside factors or conditions imposed upon it. • P(A): represents the probability P for some event A. • probability limits: For any event A, it must be that P(A) is between 0 and 1 inclusive. • probabilities of certain or impossible events: An event B certain to occur has P(B) = 1, and an event C that is impossible has P(C) = 0. MAT 312

15. complementary events: two events whose probabilities sum to 1 and that share no common outcomes. If A and B are complementary events, then P(A) + P(B) = 1. • mutually exclusive events: two events that share no outcomes. If events C and D are mutually exclusive, then P(C or D) = P(C) + P(D) If two events are not mutually exclusive, then P(C or D) = P(C) + P(D) − P(C and D). • independent events: two events whose outcomes have no influence on each other. If E and F are independent events, then P(E and F) = P(E) * P(F) MAT 312

16. conditional probability: the determination of the probability of an event taking into account that some condition may affect the outcomes to be considered. The symbol P(A|B) represents the conditional probability of event A given that event B has occurred. Conditional probability is calculated as P(A|B) = P(A and B)/P(B) • geometrical probability: the determination of probability based on the use of a 1-, 2-, or 3-dimensional geometric model. MAT 312

17. Assignment #9 (due Thursday 4/24/14) (C) Fritz wants to collect all 6 superhero figures contained in boxes of Whamo!cereal. Each specially marked box of cereal contains one superhero, randomly distributed into those boxes. Create, describe, execute, and summarize the results for a simulation to answer the following question. For consistency, we each should conduct 20 trials. How many boxes will Fritz have to purchase to collect a full set of 6 superheros? Provide simulation details as modeled in class using our 5-step simulation process. MAT 312