1. Beginning Probability
11/10/2017

2. Let’s begin with an Activity (page 282)
Back story: A soda company is running a promotion in which (it claims) 1 out of every 6 bottles wins a prize
When 7 friends go to a store and buy one soda each, the clerk is surprised when 3 out of the 7 win a prize
We are going to test whether it is reasonable to expect that 3 out of the 7 friends could have won a prize

3. Activity Individually, roll the die seven times to imitate the process
A 6 is equivalent to winning
A 1-5 is equivalent to not winning
Count how many “wins” you get

4. Activity Individually, roll the die seven times to imitate the process
A 6 is equivalent to winning
A 1-5 is equivalent to not winning
Count how many “wins” you get
Now repeat the process 4 more times (for a total of five times)
Each time, record the number of wins
Record each number of wins on the board
So if you got three wins twice, seven wins twice, and zero wins once, you would put two dots above 2, two dots above 7, and one dot above 0

5. Activity Individually, roll the die seven times to imitate the process
A 6 is equivalent to winning
A 1-5 is equivalent to not winning
Count how many “wins” you get
Now repeat the process 4 more times
Each time, record the number of wins
Record each number of wins on the board
What percentage of the time did at least three people win?
Should the clerk have been suspicious?

6. Beginning Probability
This is probably what you thought of when you chose to take AP Statistics
We’re going to start slowly:
In this section (5.1 in your textbook), we will learn to
Describe the idea of probability
Describe myths about randomness
Design and perform simulations

7. The Idea of Probability
Most behaviors and events occur due to some degree of chance, or randomness
Even the Browns would beat the Patriots sometimes if they played enough times
If we were able to run the 2016 presidential election numerous times, both candidates would win some percentage of the times
Of course, the election happens only once
Even though I have always been here on time to class every day, there is some chance (however small) that I will be late to class on any given day

8. Probability
Most statisticians agree that most things in life are probabilistic
There is very rarely a 100% chance of anything, and very rarely a 0% chance of anything
We can get very very close to the point that it is, for all practical purposes, a guarantee
Such as the probability that the sun will rise tomorrow (very likely)
Or the probability that Elizabeth, CO will experience a terrorist attack tomorrow (very unlikely)
We are usually going to deal with events that are somewhere in the middle

9. The Idea of Probability (Long Run vs Short Run)
All probabilistic behavior is unpredictable in the short run
If there is a 39% chance that there would be a quiz next class, you would not be very confident about whether you needed to study
But if I said that each day in November (including weekends) had a 39% chance of having a quiz, would you expect there to be 28 quizzes in the month?

10. The Idea of Probability (Long Run vs Short Run)
But if I said that each day in November (including weekends) had a 39% chance of having a quiz, would you expect there to be 28 quizzes in the month?
NO!
You intuitively know that, in the long run, chance behavior IS predictable
This is due to the law of large numbers
If we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value

11. Probability (Definition)
The probability of any outcome of a probabilistic process is a number between 0 (never occurs) and 1 (always occurs) that describes the proportion of times that the outcome would occur in a very long series of repetitions
So a probability of .5 means that, in the long run, the event will occur half of the time
How long is the long run?
50 repetitions? 100? 1000? 1 million?
The answer is “infinitely many repetitions”

12. Infinitely Many Observations
We could flip a coin 1 million times
And we could feasibly get 500,042 times coming up heads, rather than 500,000
It is very close to .5, but not quite exact
We can always flip more…and more…so we can never actually get to infinity
As the number of repetitions gets larger, the closer and closer to the true value we will get (on average)

13. Activity #2 Create a graph like the one on the board
Then flip a coin 50 times
In our case, using your die, 1-3 is “heads” and 4-6 is “tails”
So roll your die 50 times
After EACH flip, recalculate and graph the proportion of your flips that were heads
So if your first one is heads, you would be at 1
If your second one was tails, you would then be at .5
If your third one is also tails, you would now be at .33
And so on
Connect the dots

14. Activity #2
published.macmillanusa.com/stats_applet/stats_applet_10_ prob.html
As the number of repetitions gets larger, the variability (or ‘noise’) decreases
We consistently hover around our true value (after a large enough number of repetitions)

15. Myths about Randomness
The myth of short-run regularity
The idea that because randomness is predictable in the long- run, it is also predictable in the short run
Just because a coin WILL come up heads about 50% of the time in a large number of repetitions, people often believe that it should come up about 50% in a small number of repetitions
It would not be all that surprising to flip a coin three times and get three tails
A 12.5% chance (we’ll learn this later)

16. The myth of short-run regularity
In 2008, New York Times writers decided to see how remarkable, or unexpected, Joe Dimaggio’s 56 game hitting streak is
They took the actual season-long performance of every major league baseball player in every season in baseball history
They ran a simulation from these data (we’ll talk more about simulation in a few minutes)
Simulated each season 10,000 times
So in each of these 10,000 simulated “histories of baseball,” about 42% of them had a hitting streak at least as long as 56 games
56 games was extremely long and extremely unlikely for any individual to complete at any given time, but not very unlikely for SOMEONE to complete in the long-run

17. The Myth of the “Law of Averages”
If I flip a fair coin 7 times, and get 7 tails, is the next flip more likely to be heads or tails?

18. The Myth of the “Law of Averages”
If I flip a fair coin 10 times, and get 10 tails, is the next flip more likely to be heads or tails?
Correct answer: It is a 50% chance of both
The “Law of averages” is often used to say that everything will even out, so if we have had 10 tails in a row, the next one MUST be a heads
Getting 11 tails in a row is extremely unlikely, but once we have already had 10 tails in row, it is still just a chance whether we get an 11th
Each repetition is independent
This is true even in athletics, where athletes are often said to be “due” if they are having a bad day
If a basketball player that normally makes 40% of his shots has missed 8 in a row, the chance that he makes the next one still seems to be 40%

19. The Myth of the “Law of Averages”
A few years ago, a couple planned to have 4 kids. But the first 4 they had were girls. They wanted at least one boy
Once they had had 7 girls, they asked the doctor if it made sense to keep trying
The doctor said yes, the law of averages was on there side, and that the next one was overwhelmingly likely to be a boy
They had an 8th girl—a big surprise to them, but not to statisticians!
The “Law of Averages” is not a real thing
It does not exist
It comes from a misunderstanding of the law of large numbers

20. Simulations We have already started doing simulations in this class
The activity earlier in the year where we tried to find what percentage of a set of cards was red vs. black
The activity earlier in the year where we tried to find whether the airline was discriminating against male pilots
The activities today using dice
A simulation is the imitation of chance/probabilistic behavior using a model that accurately reflects the situation

21. Performing Simulations
What is the question we are trying to answer?
Describe how we are going to set up the simulation to answer this question
Perform many repetitions of the simulation
Use the results of the simulation to answer the question of interest
Your answer should be an approximation
So if you run a simulation in which the event happens 18 times out of 100
You probably want to conclude that the probability is approximately .18, instead of saying that the probability is .18

22. Performing Simulations
So let’s do this for our dice simulation earlier today
State our question:

23. Interpreting Probabilities
When drug testing athletes, there is always some small chance of a “false positive” (they test positive for the drug even though they haven’t taken it) and a small chance of a “false negative” (they test negative even though they DID take the drug. For one particular drug, the probability of a false negative is 0.03.
A. interpret this probability

24. Interpreting Probabilities
When drug testing athletes, there is always some small chance of a “false positive” (they test positive for the drug even though they haven’t taken it) and a small chance of a “false negative” (they test negative even though they DID take the drug. For one particular drug test, the probability of a false positive is 0.03.
A. interpret this probability
If we tested a very large number of athletes who had not taken performance-enhancing drugs, about 3% of the tests would say that the athlete actually HAD taken the drug

25. Performing Simulations
So let’s do this for our dice simulation earlier today
State our question: What is the probability that 3 out of 7 people win bottle-cap prizes if each bottle has a 1/6 chance of winning?
Describe how: Use a six-sided die to simulate the outcome for each person. A 6 on the die means he/she wins, while a 1-5 means he/she doesn’t win.
Repeat this process for a total of 5 repetitions per person (for a total of more than 50 repetitions for the class)
Based on our simulations, 3 people out of 7 would win with probability approximately ________

26. Simulation Examples
football/predictions?season=2017&seasonType=1&week=11&gr oup=-3
forecast/?ex_cid=rrpromo#plus
predictions/?ex_cid=rrpromo
ten-million-simulations-tell-us-about-president-trumps

27. Useful Tools
On your homework, it may ask you to use Table D in your book to select random numbers
A reminder that randInt(low,high) on your calculator will do random numbers too
Random.org will do all sorts of stuff
Flipping coins from centuries ago
Rolling virtual dice
Picking cards from a deck