1. Chapter 5 5.1 Randomness, Probability, and Simulation
Outcome: I will interpret probability as a long-run relative frequency and I will use simulation to model chance behavior.

2. Video to watch!
Probability Video

3. Why do the rules of football require a coin toss?
Tossing a coin seems like a “fair” way to decide who gets the ball first.
One reason why statisticians recommend random samples/randomized experiments is that they avoid bias by letting chance decide who gets selected or who receives which treatment.
Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run
This fact is the basis for the idea of probability.

4. 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.
We call this value probability.
For example: We toss a coin to see if we get heads or tails. If we toss it enough times, the proportion of heads in many tosses eventually closes in on 0.5

5. Probability
The probability of any outcome of a chance process is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions.
It is the predicted long-run relative frequency
Outcomes that never occur have a probability of 0. Outcomes that happen on every repetition ha a probability of 1.

6. Activity
Pretend you are flipping a fair coin. Without actually flipping a coin, imagine the first toss. Write down the result you see in your mind: heads (H) or tails (T).
Imagine a second coin flip. Write down the result.
Keep doing this until you have recorded the results of 50 imaginary flips. Write your results in groups of 5 to make them easier to read
Ex. HHTTT TTHTH, etc.

7. A run is a repetition of the same result
A run is a repetition of the same result. In the example in Step 3, there is a run of 2 tails followed by a run of 3 heads: (TTHHH). Read through your imaginary coin flips and find the longest run.
Record your longest run on the dotplot on the board.
Now, take a coin and flip it 50 times to generate a similar list. Find your longest run.
Record your longest run on the other dotplot on the board.
Compare distributions of longest run from imagined tosses and random tosses. What do you notice?

8. Check Your Understanding:
P. 292, The Practice of Statistics 5th Ed.

9. Myth of randomness
You can see some interesting behavior in a casino. When the shooter in the dice game called craps rolls several winners in a row, some gamblers think she has a “hot hand” and bet that she will keep on winning.
Others say that the “law of averages” means that she must now lose so that the wins and losses will balance out. Believers in the “law of averages” think that if you toss a coin 6 times and get TTTTTT, the next toss will most likely be a head.
It is true that in the long run, heads will appear about half of the time. What is a myth is that future outcomes must make up for an imbalance like six straight tails.

10. Coins and dice have no memory
Coins and dice have no memory. A coin doesn’t know that the first 6 outcomes were tails and it can’t try to get a head on the next toss to even things out.
Of course, things will even out in the long run. That’s the Law of Large Numbers in action.

11. Simulation
An imitation of chance behavior, most often carried out with random numbers.
To perform a simulation, follow the four step process:
STATE: Ask a question about some chance process
PLAN: Describe how to use a chance device to imitate one repetition of the process.
DO: Perform many repetitions of the simulation.
CONCLUDE: Use the results of your simulation to answer the question of interest

12. Devices for Randomness
Slips of paper
Spinner
Dice
Coin flip
Table of random digits
Calculator random number generator
Advantages for each? Disadvantages?

13.
Simulation Activity!

14. Homework!

15. Chapter 5 5.1 Randomness, Probability, and Simulation
Outcome: I will interpret probability as a long-run relative frequency and I will use simulation to model chance behavior.