1. Lecture 6

2. Paper 1
Select an area of human endeavor (e.g., a particular sport, marketing, politics, car manufacturing …) and write a 5 page paper on how statistics influenced it.
Due on Thursday 9/29.
Make sure you clearly reference your sources (including websites)

3. Probability: the study of randomness
Randomness Basic Probability models – language Simulations

4. Randomness Which is random?
A lot more “structure than people give credit”

5. The language of probability
In statistics, random means more than just unpredictable or haphazard.
A random phenomenon is a situation in which
the outcome is uncertain, but
there would be a definite distribution of outcomes if the situation were repeated many times under identical conditions. (And the same distribution would result if it were repeated many times again.)

6. The language of probability
Examples
Toss a coin, note whether it comes up H or T
Take a SRS from a population, find proportion who call themselves Unaffiliated
Choose 40 days at random from the 365 days of the year, check whether any day is repeated
Not a random phenomenon: UNC-Duke game, …
In each example but the last, we can ask: what would we expect for the distribution of outcomes after many repetitions? (That’s what makes them random phenomena.)

7. Probability is expected long-run proportion
Results of a simulated series of 10,000 tosses of a fair coin
tosses
results so far
expected results so far
discrepancy (observed – expected)
proportion of heads
1-10
8 H, 2 T
5 H, 5 T
+3 H .8
11-50
23 H, 27 T
25 H, 25 T
– 2 H
.46
51-100
49 H, 51 T
50 H, 50 T
– 1 H
.49
248 H, 252 T
250 H, 250 T
.466
475 H, 525 T
500 H, 500 T
– 25 H
.4750
4952 H, 5048 T
5000 H, 5000 T
– 48 H
.4952
The “law of averages” applies to the proportion of heads, not the numbers of heads and tails.
It doesn’t operate by making up a discrepancy, but by swamping it.

8. R code
n=10000 p=.5 set.seed(2013) u=runif(n) x=(u>p) y=cumsum(x) print(y[c(10,50,100,500,1000,10000)]) print(c(10,50,100,500,1000,10000)-y[c(10,50,100,500,1000,10000)]) print(y[c(10,50,100,500,1000,10000)]-c(10,50,100,500,1000,10000)/2) print(y[c(10,50,100,500,1000,10000)]/c(10,50,100,500,1000,10000)) #PLOT THE RESULTS par(mfrow=c(2,2)) plot(1:30,x[1:30]) plot(1:n,y,type='l') plot(1:n,2*y-1:n,type='l') lines(c(0,n),c(0,0),lty=2) plot(1:n,y/1:n,type='l') lines(c(0,n),c(p,p),lty=2)

9. Plotting in R Commands plot(x,y,type=“l”) lines(x,y) points(x,y)
Creates a new plot (without the type=“l” plots dots)
lines(x,y)
points(x,y)
Adds additional plot without erasing the old one

10. Probability is expected long-run proportion
The “law of averages” applies to the proportion of heads, not the numbers of heads and tails.
It doesn’t operate by making up a discrepancy, but by swamping it.
Illustration to follow: a simulated coin-tossing game. On each toss, you win $1 if the coin lands heads, and you lose $1 if it lands tails.
So, after a number of tosses, your net gain is
number of heads – number of tails.
(Net loss if this is negative.)
Should this be near zero after many tosses? Not necessarily.
The proportions of heads and tails will be near .5, but that doesn’t mean the difference between numbers of heads and tails will be near zero.
Use cointossdemo2

11. Subjective probability
The probability based on repeated sequence is objective – independent on a user.
Probability that UNC will win ACC title.
Not objective – only one observation.
We all have our own beliefs about that event.
The probability can be evaluated using bets.
If I am willing to bet 1:9 on UNC (win $9 if UNC wins, loose $1 if they lose) my probability is >10%,
If I am willing to offer 1:9 bet on UNC (loose $9 if UNC wins, win $1 if they lose) – my probability is <10%
Can be objectified for an individual using a series of bets.
The fact that people play lottery/casino is a proof that we are poor judges of subjective probabilities. This is especially true for small probability events.

12. Betting sights
Trade on future outcome – at expiry 100 if happens/ 0 if it does not
In the past – intrade.com (closed by regulators)