Chapter Probability. Probability is the branch of mathematics that describes the pattern of chance outcomes

Transcription

1 Chapter Probability Probability is the branch of mathematics that describes the pattern of chance outcomes

2 As a special promotion for its 20 oz bottles of soda, a soft drink company printed a message on the inside of each bottle cap. Some of the caps said, Please try again! while others said, You re a winner! The company advertised the promotion with the slogan 1 in 6 wins a prize. The prize is a free 20 oz bottle of soda, which comes out of the store owner s profits. Seven friends each buy one 20 oz bottle at a local convenience store. The store clerk is surprised when 3 of them win. The store owner is concerned about losing money from giving away too many free sodas. She wonders if this group of friends is just lucky or if the company s 1-in-6 claim is inaccurate. For now, let s assume that the company is telling the truth, and that every 20 oz bottle of soda as a 1-in-6 chance of being a winner. We can model the status of an individual bottle with a six-sided die: let 1 through 5 represent Please try again! and 6 represents You re a winner!. Roll your die seven times to imitate the process of the seven friends buying their sodas. How many won a prize? Add your number of wins to the dotplot on the board. Do this simulation one more time. How many won a prize?

3 The Idea of Probability Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run. The law of large numbers says that if we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value. Definition: The probability of any outcome of a chance process is a number between 0 (never occurs) and 1(always occurs) that describes the proportion of times the outcome would occur in a very long series of repetitions. Probability is long-term relative frequency

4 Pretend you are flipping a coin 30 times. Write down the results you see in your mind. A run is a repetition of the same result. Plot the length of the longest run you had. Now use your calculator to simulate these flips. Be sure you "seed" your calculator. This will insure that the same random numbers do not appear on everyone's calculator. Enter a random number in the calculator such as the student ID number or telephone number. Then press [STO->] [MATH] "PRB" "1:rand. We want to pick 30 numbers, either 1 (let this represent tails) and 2 (heads). RandInt(1,2,30)" [ENTER]. What is the length of your longest run?

5 Myths About Randomness The idea of probability seems straightforward. However, there are several myths of chance behavior we must address. The myth of short-run regularity: The idea of probability is that randomness is predictable in the long run. Our intuition tries to tell us random phenomena should also be predictable in the short run. However, probability does not allow us to make short-run predictions. The myth of the law of averages : Probability tells us random behavior evens out in the long run. Future outcomes are not affected by past behavior. That is, past outcomes do not influence the likelihood of individual outcomes occurring in the future.

6 Thinking about randomness You must have a long series of independent trials The outcome of one trial must not influence the outcome of any other Empirical Probability the probability of simulations Theoretical Probability the probability of mathematical formulas Computer simulations are very useful because we need long runs of trials Short runs give only rough estimates of a probability

7 Simulation The imitation of chance behavior, based on a model that accurately reflects the situation, is called a simulation. Performing a Simulation State: Ask a question of interest about some chance process. Plan: Describe how to use a chance device to imitate one repetition of the process. Tell what you will record at the end of each repetition. Do: Perform many repetitions of the simulation. Conclude: Use the results of your simulation to answer the question of interest. We can use physical devices, random numbers (e.g. Table D), and technology to perform simulations.

8 Simulation is an effective tool for finding likelihoods of complex results once we have a trustworthy model Table of random digits Graphing calculator Computer software EX: In an attempt to increase sales, a breakfast cereal company decides to offer a NASCAR promotion. Each box of cereal will contain a collectible card featuring one of these NASCAR drivers: Jeff Gordon, Dale Earnhardt Jr., Tony Stewart, Danica Patrick, or Jimmie Johnson. The company says that each of the 5 cards is equally likely to appear in any box of cereal. A NASCAR fan decides to keep buying boxes of the cereal until she has all 5 drivers cards. She is surprised when it takes her 23 boxes to get the full set of cards. Should she be surprised? Design and carry out a simulation to help answer this question. 8

9 1. State Simulation basics I must have all 5 of those NASCAR cards! We will model picking boxes of cereal from our random digit table with one of the 5 NASCAR cards in it. 2. Plan Probability of picking each card is 1/5 so we need five numbers to represent the five possible cards. Let s let 1 = Jeff Gordon, 2 = Dale Earnhardt, Jr., 3 = Tony Stewart, 4 = Danica Patrick, and 5 = Jimmie Johnson. 9

10 Using the RNT 3. Do Select a line in the RNT (let s use line 115) We only want to look at numbers between 1 and Conclude We had to buy 14 boxes of cereal to get all 5 cards. So our estimate of the probability that it takes 23 or more boxes to get a full set is very low (roughly 0). That NASCAR fan should be surprised about how many boxes she had to buy.

11 Assigning Digits Consider a random group of people in which 70% are employed Consider a random group of people in which 73% are employed 0, 1, 2, 3, 4, 5, 6 = employed 7, 8, 9 = not employed Now you have to use 2 digit numbers (don t use them unless you really have to!) 00, 01, 02, 03,, 72 = employed 73, 74, 75,, 99 = not employed 11

12 Suppose a cereal manufacturer puts cards of famous athletes in boxes of cereal in the hope of boosting sales. The manufacturer announces that 20% of the boxes contain a Alex Rodriguez card, 30% contain a card of Michael Phelps and the rest contain a card of Serena Williams. You must have all three cards! How many boxes of cereal do you expect to have to buy in order to get the complete set?

13 1. State I want all 3 of those cards so I want to simulate picking cereal boxes of these probabilities: A-Rod= 20%, Phelps = 30%, Williams = 50% 2. Plan Assign digits to represent outcomes A-Rod = 0, 1 Phelps = 2, 3,4 Williams = 5, 6, 7, 8, 9 4. Do Use line 150 in RNT to perform simulation 13

14 3. Do This means the order of cards I get is: AR, SW, SW, AR, AR, SW, SW, SW, AR, SW, MP 4. Conclude I had to buy 11 boxes of cereal before I had all 3.