1. + The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 5: Probability: What are the Chances? Section 5.1 Randomness, Probability, and Simulation

2. + 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 aspecific 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.

3. + A Common Question The probability of tossing a coin and it landing on Heads is 0.5. Theoretically, then, if I toss the coin 10 times, I should get 5 Heads. However, with such a small number of tosses there is a lot of room for variability. There are two games involving flipping a coin. Game 1: You win if you throw 40% - 60% heads. Game 2: You win if you throw more than 75% heads. For which game would you rather toss the coin 50 times? 500 times?

4. + Simulations What are they? Imitations of an experiment – a way to calculate the probability of something happening without actually carrying out the experiment Why do we use them? It can be quicker and/or less expensive than actually carrying out the experiment. The laws of probability can be confusing. Simulation makes sense.

5. + Example Suppose I want to know the probability that a couple has a girl among their first four children. Assume that: The probability of having a boy = the probability of having a girl = 0.5 What real life object simulates 2 outcomes of equal likelihood?

6. + Example Continued So, we could designate HEADS = having a girl. Simulation: Toss the coin 4 times and record whether HEADS shows up at least once. Or, we could use a well-shuffled deck of cards. Let red = having a girl and black = having a boy. Choose 4 cards and record whether a red card shows up at least one. Or, we could use a table of random digits. Let even = having a girl and odd = having a boy. Choose 4 single digits. Record whether an even number shows up at least once. Repeat this many times and compute the probability.

7. + Some more examples Shaq’s forte is definitely not on the free throw line. Let’s assume he is a 60% free throw shooter. How could we simulate an experiment to see how many shots he would have to take to make 5 in a row? How would this problem change if Shaq is a 47% free throw shooter?

8. + Synopsis of the Steps in a Simulation (State, Plan, Do, Conclude) What is the question of interest about some chance process? Explicitly detail how the simulation will be carried out. What will the digits represent? What digits are not used? How many digits will be chosen at a time from the table of random digits? How will you know when to stop? What will you count? Perform many repetitions of the simulation. Use the results of your simulation to answer the question of interest.

9. + Example: Golden Ticket Parking Lottery Read the example on page 290.What is the probability that a fair lottery would result in two winners from the AP Statistics class? StudentsLabels AP Statistics Class01-28 Other29-95 Skip numbers from 96-00 Reading across row 139 in Table D, look at pairs of digits until you see two different labels from 01- 95. Record whether or not both winners are members of the AP Statistics Class. 55 | 5889 | 9404 | 7070 | 8410|98|4356 | 3569 | 3448 | 3945 | 17 X | X ✓ | X X | X ✓ |Sk|X X | X X | ✓ No 19 | 1297|51|3258 | 1304 | 8451 | 4472 | 3218 | 1940|00|3600|24|28 ✓ | ✓ Sk|X|X X | ✓✓ | X X | X ✓ | ✓ X|Sk|X Sk| ✓ | ✓ YesNo YesNoYes Based on 18 repetitions of our simulation, both winners came from the AP Statistics class 3 times, so the probability is estimated as 16.67%.

10. + Example: NASCAR Cards and Cereal Boxes Read the example on page 291.What is the probability that it will take 23 or more boxes to get a full set of 5 NASCAR collectible cards? DriverLabel Jeff Gordon1 Dale Earnhardt, Jr.2 Tony Stewart3 Danica Patrick4 Jimmie Johnson5 Use randInt(1,5) to simulate buying one box of cereal and looking at which card is inside. Keep pressing Enter until we get all five of the labels from 1 to 5. Record the number of boxes we had to open. We never had to buy more than 22 boxes to get the full set of cards in 50 repetitions of our simulation. Our estimate of the probability that it takes 23 or more boxes to get a full set is roughly 0. 3 5 2 1 5 2 3 5 4 9 boxes 4 3 5 3 5 1 1 1 5 3 1 5 4 5 2 15 boxes 5 5 5 2 4 1 2 1 5 3 10 boxes

11. + Past AP Problem 2001 #3