# Probability

Lake Wobegon -- "where the women are strong, the men are good-looking, and all the children are above average."
--Garrison Keillor, A Prairie Home Companion

The average person has an above average number of legs.
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Many people believe in ESP and other paranormal phenomena because they have a poor understanding of probability.

## Introduction to Probability

http://www.mathgoodies.com/lessons/vol6/intro_probability.html

e.g. The probability of rolling a 2 on a fair die is P(2)=1/6. One favorable outcome out of six possibilities.

The probability of rolling three 2s in a row (a 2 AND another 2 AND another 2) is P(2)*P(2)*P(2) = 1/6 * 1/6 * 1/6 = 1/216. When events depend on each other (connected by the word "AND") then multiply the individual probabilities.

The probability of rolling an even number on one roll of the die (the probability of rolling a 2 OR a 4 OR a 6) is P(2)+P(4)+P(6) = 1/6 + 1/6 + 1/6 = 1/2. When events do not depend on each other (connected by the word "OR") then add the individual probabilities.

The probability of an event (X) NOT to occur is 1-P(X).

Poker hands are arranged in order of increasing order of probability.
e.g. A full house beats a flush because a full house is LESS likely to occur.
http://mathforum.org/library/drmath/view/56616.html

## Gambler's Fallacy

• EXAMPLE: which of the following number sequences was generated by a random process (multiple are possible)
• (a)
`8 6 9 0 6 9 7 9 2 3 9 9 7 6 9 9 5 4 4 1`

• (b)
`6 2 1 2 3 9 4 6 8 1 9 3 2 5 0 7 3 8 1 5  `

• (c)
`1 9 7 7 7 4 3 4 4 8 4 6 8 1 3 9 4 5 0 1`

• (d)
`7 3 5 5 0 2 7 7 8 6 5 4 5 6 9 3 3 3 9 6`

• (e)
`9 2 2 2 1 9 2 4 8 2 2 9 0 7 3 3 4 8 7 5`

To see the answer, left-click on your mouse and highlight the next blank region (it contains hidden text!):
The answer is: all except (b). Even though there are apparent "hot streaks" of the same number in (a) and (c-e), those were generated by a very good random number generator. There is no dependence whatsoever on the outcome of the previous number. In (b), however, the numbers are intentionally chosen to avoid repeats and sequences.

The idea is for everyone to call out their birthday (one at a time, of course) and have anyone with the same birthday raise their hand. If one of the profs bet that we would get at least one match, how many would take the other side of that bet?

## Coin Flip Exercise

This is a simple experiment. Everyone will be given a form to use for recording results. It has 2 identical parts made up of 100 squares for recording the results of a coin toss (real or imagined). Everyone will flip a coin once to determine which part (top or bottom) of the form to use for the "brain" sequence. Record this choice in your notebook and do not write it on the form. Next - using your imagination - generate a random sequence of heads/tails (1/0) in the 100 boxes. This is the "brain" sequence. When done, flip a real coin 100 times and record the heads/tails results (as 1/0) in the other part of the form. This is the "coin" sequence. Professors Cotton and Scalise will attempt to determine which is which.

We also checked the maximum length of runs of heads or tails for everybody. Here are the results.
```
Run
Length        Occurrences
1                 x
2                 x
3                 x
4                 x
5                 x
6                 x
7                 x
8                 x
9                 x
10                x
11                x
12                x
>12               x
```
Notice the peak at x.

## Telephone game

We played the old game of "Telephone." A nonsense story starts at one end of the room and is passed on one person at a time. The results at the end are compared to the original story.

Here's our original story:

The stories are really ridiculous.
Wait till you see the next one.
We'll add this term's
results after we do the experiment.

The front of the room turned it into:

Front
results

The back half of the room made it into:

Back
results

Moral: stories passed through many brains get mangled. Be careful about stories passed by word-of-mouth. And don't sit in the back of the room!

## Clustering Illusion

The dots are distributed randomly in two dimensions, but your brain will find patterns in the randomness that do not really exist. Play with this one a bit. Try using 2000 dots. Notice what the clusters and voids do each time you run it.

Sometimes two or more studies can individually support one conclusion, but the combined statistics support the opposite conclusion.

If A is better than B and B is better than C, then how is A related to C? Surprisingly, A is not always better than C! Remember the kid's game Rock-Paper-Scissors: Rock breaks Scissors, Scissors cut Paper, but Paper covers Rock.

## Extrapolation

Extrapolation is an attempt to predict some phenomenon that lies outside the basis of experience. We looked at a table of record times for running the mile. Since 1913 there has been a steady downward trend. Prof. Scalise has plotted the times against the year, and from that we can see a roughly linear function. Now for the fun. We extrapolate and extend this linear function into the future and see that, in about 2050, someone will run the mile in zero seconds! Obviously, the extrapolation is not valid.

An extrapolation figured into analysis of the foam strike that resulted in the destruction of the shuttle Columbia in 2003. Data about the piece of foam that was observed to strike Columbia were fed into the "crater" model that NASA engineers used to evaluate the effect of foam strikes. Given the size of the piece and the impact velocity, the model would return a damage value. When parameters for the observed foam strike were fed to crater, it indicated that, while the damage would be significant, it was not a real hazard. This was an extrapolation, as the piece that hit Columbia was 400 times larger than any piece ever seen. Operating outside of the experience base, the model returned an incorrect estimate.

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