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
Many people believe in ESP and other paranormal phenomena because they have
a poor understanding of probability.
Class Statistics
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
Birthday Paradox
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?
We had about 56 people in the room. Professor Scalise had everybody
call out their birthday, one at a time. The idea was to see if any two
people in the room had the same birthday. We found a total of 5 pairs
of people sharing birthdays.
How many people are required to make the probability of this reach 0.5 or
50%? Surprisingly, when you have 23 people, the probability is
0.507.
Coin Flip Exercise
Each student flipped a coin 100 times and recorded 1 for heads and 0
for tails. Everybody also imaginatively generated a "random" sequence
of 1s and 0s in the other half of the form. Everyone was to choose
randomly which half of the form to use for the sequences and keep a note
of that. 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 0
2 0
3 0
4 2
5 15
6 13
7 8
8 3
9 3
10 1
11 1
12 0
13 1
Notice the peak at 5 to 6.
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:
Four squirrels and a tomcat went downtown for a pizza.
The anchovies smelled like roses and the wall was decorated with old socks.
The front of the room turned it into:
Forest squirrels eat pizza.
The back half of the room turned it into:
Four squirrels ate a pizza and it was great.
Moral: stories passed through many brains get mangled. Be careful about
stories passed by word-of-mouth. And don't sit in the front 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.
Simpson's Paradox
Sometimes two or more studies can individually support one conclusion,
but the combined statistics support the opposite conclusion.
Non-transitive Paradox
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.
Coincidence
Outline