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 xx 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 y 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 3
5 6
6 10
7 9
8 9
9 7
10 1
11 1
12 or more 0
Our two psychic profs did it again! In attempting to guess which of your
sequences was the "brain" sequence and which was the coin, Prof Scalise
got all right and missed none! Random guessing would be expected to
produce about 30 or so right and 30 wrong. Not bad!
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:
"A dragon and two and a half dinosaurs ate pizza in a gymnasium. Jack Daniels fell over
and polluted the crabgrass."
The front of the room turned it into:
"The dragon eats people and drove away."
The back half of the room turned it into:
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.
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.
Coincidence