1. 5.3: Simulation NEW SEATING CHARTS NEW WEBSITE FOR NOTES

2. Pick a Partner! Partner #1: You will pick a random whole number from 1 to 5 twice and add them. Partner #2: You will pick one random whole number from 1 to 10. High score wins. What would you guess is the proportion of times Partner #1 would win? Use the calculator and play as many rounds as you can in 2 minutes. Keep track of wins and losses. Enter the following commands at the home screen. randInt(1,5,200) L1: randInt(1,5,200) L2: randInt(1,10,200) L3: ((L1+L2) > L3) L4: sum(L4) / 200 (you can find “sum” in CATALOG; “>” in TEST; and “:” by hitting ALPHA and the decimal point buttons) Enter class results in L4. Define a viewing window of X(0.4 to 0.6 with a scale of 0.025 and Y(-4 to 14 with a scale of.1). Then, plot a histogram with a boxplot superimposed. Use TRACE when observing the histogram. Then 1-Var Stats L4. Do the results match your predications?

3. Introduction to Probability In a few weeks we will be able to answer the following: If we know the blood types of a man and a woman, what can we say about the blood types of their future children? Give a test for the AIDS virus to the employees of a small company. What is the chance of at least one positive test if all the people tested are free of the virus? We will begin by asking…

4. The 3 Methods If you flip a coin ten times, what is the likelihood of three or more consecutive heads or tails? A couple plans to have children until they have a girl or until they have four children – whichever comes first. What are the chances that they have a girl? We can answer these questions using one of three methods: 1. Complete many experiments and calculate the relative frequency. Time and financial constraints may limit us here. 2. Probability Models that calculate theoretical answers (Next Chapter) 3. Start with an experiment model. Develop a procedure to imitate, or simulate, a number of repetitions.

5. Three in a Row? Simulation – imitation of chance behavior, based on a model that reflects the experiment under consideration. Simulation #1: Three in a Row? 1. State the problem or describe the experiment: Toss a coin ten times. What is the likelihood of a run of at least three consecutive heads or tails? 2. State the assumptions: A head or tail is equally likely. Tosses are independent of each other.

6. Three in a Row? 3. Assign digits to represent outcomes. Table B features independent, successive digits (0-9). One digit simulates one coin toss. Odd = Heads; Even = Tails 4. Simulate many repetitions Simulate one repetition with 10 consecutive digits in Table B. Every student picks a different row and does 4 simulations. For example, Row 150:  07511 88915 41267 16853 … becomes  0751188915 4126716853 … (1 Yes and 1 No) Add the class “yes” totals and “total” simulations

7. Three in a Row? 5. State your conclusions. We estimate the probability of a run by the proportion. Estimated probability = Yes/Total = ______ A very long simulation (w/ a computer) found the true probability to be 0.826. While not the most exciting example, it was typical of many probability problems because it consisted of independent trials (tosses) with the same possible outcomes. Independent – the result of one trial has no effect on the next trial.

8. A Girl or Four? Simulation #2: A Girl or Four? A couple plans to have children until they have a girl or until they have four children, whichever comes first. 2. Each child has a probability of 0.5 of being a girl and 0.5 of being a boy. The genders of successive children are independent. 3. There are 20 blow pops in the bowl (10 red, 10 purple). Red = Girl, Purple = Boy 4. Simulate by moving around the room. Every student takes a blow pop and replaces. The number of digits needed to simulate one repetition will vary with results. 5. Conclusion: Estimated probability = 0.____

9. A Girl or Four? Simulation #2A: A Girl or Four? Now, do the same using different rows from Table B. This will allow us to do many more simulations as a class. Conclusion: Estimated Probability = 0.____ The true likelihood of having a girl is 0.938

10. Frozen Yogurt Simulation methods also work when outcomes are not equally likely. Simulation #3: Frozen Yogurt 1. Frozen yogurt sales are as follows: 38% chocolate, 42% vanilla, and 20% strawberry. The task is to simulate the next ten orders. 2. We must use two digit pairs now due to the percents. 3. 00 to 37 = Chocolate (C), 38 to 79 = Vanilla, and 80 to 99 = Strawberry 4. Select a row from Table B and use the first twenty numbers 5. How close do the next ten customers match recent history?

11. Review of Randomizing with the TI Ex) Rolling a die seven times randInt (1,6,7) Ex) Flipping a coin twenty times randInt (1,2,20)