Rock, Paper, Scissors Tournament

Transcription

1 UNIT 5: PROBABILITY Introduction to The Mathematics of Chance Rock, Paper, Scissors Tournament History As old as civilization. Egyptians used a small mammal bone as a 4 sided die (500 BC) Games of chance were common in Greek & Roman times Formal study began with Blaise Pascal & Pierre de Fermat due to gambling problems. Soda Promotion As a special promotion for its 20-ounce 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 in 6 wins a prize. Seven friends each buy one 20-ounce bottle at a local store. The store clerk is surprised when of them win a prize. Is this group of friends just lucky, or is the company s -in-6 claim inaccurate? Let s do a simulation to check this out. Assume that the company is telling the truth, and that every 20-ounce bottle has a -in-6 chance of winning We can model the status of an individual bottle with a 6-sided die: let through 5 represent Please Try Again! and 6 represent You re a Winner! Activity ( in 6). Roll your die seven times to imitate the process of the seven friends buying their soda. How many of them won a prize. 2. Repeat step four more times. In the 5 repetitions, how many times did three or more of the group win a prize?. Combine results with your classmates. What percent of the time did the friends come away with three or more prizes, just by chance? 4. Based on your answer in step, does it seem plausible that the company is telling the truth, but that the seven friends just got lucky? Simulation is a Powerful Method for Modeling Chance Behavior! # of Wins Groups - # of Times with this many wins What is the chance that all 7 walked out winners?

2 Probability In football, a coin toss helps determine which team gets the ball first. Why do the rules of football require a coin toss? (Because tossing a coin seems a fair way to decide) This is one reason why statisticians recommend random samples and randomized experiments. They avoid bias by letting chance decide who gets selected or who receives which treatment. One big fact emerges when we watch coin tosses or the results of random sampling Probability Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run. It s the long-run relative frequency. Empirical Probability This is the relative frequency that is obtained by actually doing the experiment. Activity: Coin Toss - Graph Toss coin 20 times Observe # of Heads P(Heads) = ½ How many of you landed on Heads 0 Times? # of Heads Law of Large Numbers With more repetitions, the empirical probability gets closer to the theoretical probability. Theoretical Probability is based on what I should get (in theory).based on the sample space. More Repetitions If we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value. We call this single value the probability. 2

3 Probability - Definition It is the outcome of a chance process that is a number between 0 and that describes the proportion of times the outcome would occur in a very long series of repetitions. Outcomes that never occur Probability = 0 An outcome that happens on every repetition Probability = Life Insurance: We can t predict whether a particular person will die in the next year. But, the National Center for Health Statistics says that the proportion of men aged 20 to 24 years who die in any one year is *What percent of men (20 to 24) will die? 0.5% *The probability of a woman (20 to 24) will die is one-third that of a man. What percent of women (20 to 24) will die? ( 0.5% ) = 0.05% *Would you be surprised to learn that they charge a man (20 to 24) three times more than a woman (20 to 24)? No, because the probability of having to pay is times higher. According to the Book of Odds, the probability that a randomly selected U.S. Adult usually eats breakfast is 0.6. What does the probability 0.6 mean in this setting? This probably means that if you asked a large sample of U.S. adults whether they usually eat breakfast, about 6% of them will answer yes. Does this mean that if 00 U.S. adults are chosen at random, exactly 6 of them usually eat breakfast? No because the exact number will vary from sample to sample. Match the following probabilities with each statement This outcome is impossible it can never occur. 0 This outcome is certain. It will occur on every trial. This outcome is very unlikely, but it will occur once in a while in a long sequence of trials. 0.0 This outcome will occur more often than not. This probability is 0.6. An outcome that will occur more often than not will occur in more than 50% of trials which means a probability that is greater than This leaves us with two choices: 0.60 and The wording suggests that the event occurs often but not nearly every time. This suggests something that occurs 60% of the time. Which of the following outcomes is more probable when flipping a coin? HTHTTH TTTHHH Myth of Short-run Regularity Probability is predictable in the long run not the short run.

4 Myth of the Law of Averages Things do even out in the long run. If I tossed a coin and got TTTTTT, which of the following is more likely on the next toss? Getting Heads Simulation It is an imitation of chance behavior, based on a model that accurately reflects the situation. This is what we did in the in 6 game earlier Getting Tails At a local high school, 95 students have permission to park on campus. Each month, the student council holds a golden ticket parking lottery at a school assembly. The two lucky winners are given reserved parking spots next to the school s main entrance. Last month, the winning tickets were drawn by a student council member from the AP Stats class. When both golden tickets went to members of that same class, some people thought the lottery had been rigged. There are 28 students in the AP Stats class, all of whom are eligible to park on campus. Design and carry out a simulation to decide whether it s plausible that the lottery was carried out fairly. Let s start at line 9 of the Random number table. Let the students who can park be represented as 0 to 95 and the students in the AP stats is 0 to 28. We will simulate drawing two tickets. We will use 9 trials first then 8 trials. Example: Golden Ticket Parking Lottery What is the probability that a fair lottery would result in two winners from the AP Statistics class? Students Labels AP Statistics Class 0-28 Other Skip numbers from Reading across row 9 in Table D, look at pairs of digits until you see two different labels from Record whether or not both winners are members of the AP Statistics Class X X X X X X X Sk X X X X X X X X No No No No No No No No No Sk X X X X X X X X X Sk X Sk Yes No No No No No Yes No Yes Based on 8 repetitions of our simulation, both winners came from the AP Statistics class times, so the probability is estimated as 6.67%. Do you think the Lottery was Rigged? In our simulation of a fair lottery, both winners were in the AP Statistics class in 6.67% of the time. So about in every 6 times the student council holds the golden ticket lottery, this will happen just by chance. It seems plausible that the lottery was conducted fairly. 4

5 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 kiely 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 2 boxes to get the full set of cards. Should she be surprised? Design and carry out a simulation to help answer this question. Let -5 represent finding each of the five NASCAR drivers. Since we want a full deck, then we will go until we get all 5 cards. We will keep up with the number of boxes that we had to check in order to get all 5. Let s start with line Boxes Boxes Boxes 7 Boxes 20 Boxes 0 Boxes Should the Nascar fan be surprised that it took her 2 boxes to get a full set? In our simulation, we never had to buy more than 20 boxes to get a full set in 6 repetitions of our simulation. So, our estimate of the probability that it takes 2 or more boxes to get a full set is roughly 0. The Nascar fan should be surprised about how many boxes she had to buy. What DON T simulations tell us? Golden Ticket Lottery: We concluded drawing was done fairly. Does this mean the lottery was conducted fairly? Not necessarily. All we did was estimate that the probability of getting 2 winners from AP Statistics class was 6.7% if the drawing was fair. So, the result isn t unlikely enough to convince us that the lottery was rigged. Nascar: We concluded that the fan should be surprised. Does that mean the company didn t tell the truth about how the cards were distributed? Not necessarily. Our simulation says that it s very unlikely for someone to have to buy 2 boxes to get a full set if each card is equally likely to appear in a box of cereal. The evidence suggests that the company s statement is incorrect. It is still possible, however, that the Nascar fan was just very unlucky. Activity: Is it Fair? Homework Page 29 (,, 7, 9,, 7, 9, 2, 25) 5