. . . 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 49 pennies taken from the jar. • Note the horizontal and vertical scales on the grid here and then record that (rounded) average age using a properly scaled X. MAT 312

2. Prob & Stat (MAT 312)Dr. Day Thursday April 17, 2014 • Grab 49 Pennies at Random then Plot Average Age • Review: Probability Concepts and Calculations • Experimental Probability: Probabilities Based on Data • Theoretical Probability: Mathematical Analysis • Terms and Properties • Geometrical Probability • Simulations • Incoming: Assignment #8 • Assignment #9 • Probability Distributions: Sample Spaces & Random Variables MAT 312

3. Types of Probability Experimental probability: Determination of numerical probability through the use of existing data or simulation of a real or imagined event. Theoretical probability: Examine what could happen; use counting techniques, models, geometrical representations, and other mathematical calculations and techniques to determine all things that could happen in an experiment. Express probabilities by comparing outcomes that meet specific requirements to all possible outcomes of an experiment. MAT 312

4. Theoretical Probability What happens when two number cubes are rolled and we consider the sum of the two face-up sides? Are all outcomes equally likely? • Look at all the ways the sums could occur: all possible results of the experiment. MAT 312

5. Theoretical Probability MAT 312

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7. Theoretical Probability For each experiment: • Determine all possible outcomes. • Determine the number of different outcomes that are possible. • Determine whether each of the different outcomes is equally likely. Experiment 1: Flip a fair coin and record whether the coin lands heads up or tails up. Experiment 2: Roll a fair die and record the result and then flip a coin and record the result. Experiment 3: Three fair coins are flipped simultaneously and the head/tail result is recorded. Experiment 4: April is at the free-throw line to attempt two free throws. MAT 312

8. 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

9. 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

10. 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

11. 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

12. Geometrical Probability Geometrical probability refers to the use of geometrical representations and calculations to determine the probabilities of outcomes. The outcomes must be represented directly or indirectly through 1-, 2-, or 3-dimensional geometrical shapes. MAT 312

13. Geometrical Probability If I give you a 5-inch piece of string and ask you to make one cut in the string at some random point along its length, what is the probability one of the resulting pieces will be less than 1 inch long? MAT 312

14. Geometrical Probability Suppose we throw a dart at the board shown here. If we know the dart hits somewhere on the 18-inch square board, and that the dart has hit the board at random, what is the probability of a bull's eye? That is, what's the probability that a point chosen at random on the board is within the smallest circle? MAT 312

15. 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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