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. . . PENNIES for the AGES . . .

. . . PENNIES for the AGES . . . . Push the “Sample More Data” button on the screen and read the average age of a sample of 36 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 . .

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. . . PENNIES for the AGES . . .

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  1. . . . PENNIES for theAGES. . . • Push the “Sample More Data” button on the screen and read the average age of a sample of 36 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 3, 2014 • Grab 36 Pennies at Random then Calculateand Plot Average Age • Finish: Concepts and Principles of Counting • Combinations • Probability Concepts and Calculations • Experimental Probability: Probabilities Based on Data • Theoretical Probability: Mathematical Analysis • Terms and Properties MAT 312

  3. Permutations Set I: {a,m,r} Set II: {b,d,i,l,u} Set III: {c,e,n,t} In how many ways can the letters within just one set, from among I, II, and III, be ordered? • Multiplication Principle • Factorial Notation Permutations: ordered arrangements of items. Almost every morning or evening on the news I hear about the State of Illinois DCFS, the Department of Children and Family Services. I get confused, because our mathematics department has a committee called the Department Faculty Status Committee, or DFSC. Can you see why I'm confused? How many different 4-letter ordered arrangements exist for the set of letters {D, F, S, C}? MAT 312

  4. Permutations Extension: Consider ordered arrangements of only some of the elements in a set. If Blaise will post only four possible soda choices, how many different ordered arrangements of the four sodas are there? • Notation: P(n,r): the number of ways to arrange r objects from a set of n objects. • P(4,4) MAT 312

  5. Combinations What is the distinction between these two questions? In how many ways can a 5-card poker hand be dealt? How many different 5-card poker hands exist? Permutations: P(52,5) Combinations: Selection of objects from a set with no regard for order or arrangement. Simpler Problem: How many ways can we selectthree items from the 5-element set {A,B,C,D,E} if the order of the three items is disregarded? List the unique 3-element subsets of {A,B,C,D,E}: ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE. There are 10 such 3-element subsets. Or, relate to permutations There are P(5,3) = 60 ordered arrangements of the 5-element set into 3-element subsets. Within the 60 ordered arrangements, there are 10 groups, each with 6 arrangements that use the same 3-letter subset. That is, 60 ÷ 6 = 10 unique 3-element subsets. MAT 312

  6. Combinations Notation Generalization: The number of combinations of n items selected r at a time, where the order of selection or the arrangement of the r items is notconsidered. Note: MAT 312

  7. 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

  8. Experimental or Theoretical? Determine the probability that: • a seed for a new variety of salad tomato will actually sprout; • the Chicago Cubs will win at least 12 games in April; • the average penny age will range from 23 years to 27 years when a sample of size 49 is drawn from our penny population; • A 20-mile-per-hour head-on collision will cause an upper-body injury to the driver of a 2013 Ford Focus; • a sum of 6 will result when two number cubes are rolled and the face-up values are added; • A committee with three men will result when a committee of three is selected at random from the students in this class. MAT 312

  9. Experimental Probability Determine the probability that a seed for a new variety of salad tomato will actually sprout. Conduct an experiment: Compare number of sprouted seeds to the total number of seeds planted. Suppose 986 seeds sprouted out of 1000 seeds sown. • Use a ratio: The probability is 986/1000 that a seed will sprout. • Use a decimal fraction: The probability is 0.986 that a seed will sprout. • Use a percent: The probability is 98.6% that a seed will sprout. MAT 312

  10. Experimental Probability Determine the probability that Bryce Harper will strike out at any given plate appearance. Examine Existing Data • In his 1103 plate appearances (games played through Wednesday 4/2/14), Bryce has struck out 218 times 218/11030.197619.76% MAT 312

  11. Experimental Probability Crammits Cereal Company offers one of five cartoon surprises in each box of its BigByte Cereal. Determine the probability Eddie gets all five surprises in just the first 8 boxes his family purchases. We could actually purchase the cereal and record which type of cartoon surprise is in the box, and do this for one or more sets of 8 boxes. • Problems: costly and time consuming • Other situations, other issues Carry out a simulation: Generate outcomes to calculate experimental probabilities. Simulate the actual event and use the results of the simulation to determine experimental probabilities. MAT 312

  12. Theoretical Probability Determine the theoretical probability of a 3 appearing when a number cube is rolled. Equally Likely Outcomes Compare the one way to get a 3 to the six outcomes that are possible when the experiment is conducted. 1/6 0.167 16.7% MAT 312

  13. 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

  14. Theoretical Probability MAT 312

  15. MAT 312

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. Assignment #7 (due Tues, 4/8/14, by 8:45 pm) Textbook: Chapter 6, pp 180-181 • #3 • #4 • #5ab • #6a • #7ac • #8ab • #10abc Provide explanation and numerical response! MAT 312

  22. Assignment #8 (due Tues, 4/15/14) Textbook: Chapter 6, pp 180-182 • #6b • #7efg • #8ce • #11 • #12cdef Textbook: Chapter 5, pp 148-149 • #9 • #10efg Provide explanation and numerical response! MAT 312

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