Section 5.1 Randomness, Probability, and Simulation

Transcription

1 Section 5.1 Randomness, Probability, and Simulation What does it mean to be random? ~Individual outcomes are unknown ~In the long run some underlying set of outcomes will be equally likely (or at least the likelihood of outcomes will be consistent with actual outcomes in the long run) If someone handed you a coin and told you that it s biased toward usually landing heads, you d be justly skeptical. What would you do? 1

2 You d probably test the coin by tossing it a few times. How many times are enough? Would 3 heads in a row convince you that the coin was biased? Would 7 of 10 heads be convincing? Imagine we use 100 tosses. Would 54 heads out of 100 convince you that the coin is biased? What about 60 out of 100? How about 95 out of 100? How about 80 out of 100? or 70? 2

3 Let s decide as a class what our rule will be. Let s use a simulation to answer this question. randint(0, 1) simulating a single coin toss (with 0 = tails, 1 = heads) randint (0, 1, 5) simulating tossing 5 coins sum(randint (0, 1, 100)) since each coin is 0 or 1, the sum of these 100 values is equivalent to counting the number of heads 3

4 Let s try running a couple of these how many people got more than heads? Keep in mind that even if we come up with a rule, tossing a coin 100 times and declaring it okay or biased would still not give us the truth for sure just a suggestion that the coin is biased or not. 4

5 The steps of a simulation the basic unit of our simulation, each randomly generated digit gives the outcome of ONE component Identify the component to be repeated Explain how you will model the component s outcome a combination of component outcomes that give a simulated answer to our question Explain how you will combine the components to model a trial Run several trials if by hand ~20 is good the outcome of the trial, what we are measuring with each trial what will we use to summarize our results Conclusions - the simulation SUGGESTS... Suppose a basketball player has an 80% free throw success rate. How can we use random numbers to simulate whether or not she makes a foul shot? How many shots might she be able to make in a row without missing? Describe the simulation of having her shoot free throws until she misses, counting the number of successes. 5

6 How would the simulation change if her success rate were only 72%? How would a trial and our response variable change if we want to know how many shots she might make out of 5 chances? 6

7 How would a trial and our response variable change if we want to know her chances of hitting both shots when she goes to the line to shoot two? Random Digits Table Another way to generate random (or as random as we can get) is using a table of random digits. There is one in the back of your book. Each digit 0 through 9 is equally likely to occur at any time, therefore any pattern of two or more digits is equally likely to occur at any point in the table. In order to use this table to generate random digits - pick a row to start on. Read the digits left to right being sure to read in groups appropriate to the size of the numbers assigned for your simulation. If your numbers go from 1-15 you must read two digits at a time, etc. 7

8 Section 5.1 Notes - Completed Suppose we are to randomly select three students from the class to speak at Parents Night about taking AP Stats. How likely is it we ll get three boys? Sam is a basketball player who typically makes 85% of her free throws. However, when shooting two free throws in a row, the likelihood of her making the second shot depends upon how she does on the ]irst shot. If she misses the ]irst shot, she only makes 70% of her second free throws. But if she makes her ]irst shot, she makes 90% of her second free throws. Describe a simulation that will determine the chance of Sam making two free throws in a row. 8

9 Homework: p. 293 #s 1, 3, 7, 9, 11, 15, 17, 19, 23, 25 9