Randomness and Coincidence
Randomness and Coincidence
Proving That a Coin is Biased
We will do something really interesting  prove that a coin is biased! It's easy to do. Everyone in the class gets out a coin, a pen and paper. Everyone then flips their coin ten times and counts the heads and tails. We then count who got what combinations.
The results were interesting; we found several biased coins. Look at the results of the flipping.
H T Count 0 10 x 1 9 x 2 8 x 3 7 x 4 6 x 5 5 x 6 4 x 7 3 x 8 2 x 9 1 x 10 0 x
The Gambler's Fallacy
Suppose you tossed a coin and got 8 consecutive tails. What would be the probability of getting a tail the next time? Answer: 0.5. Just as in all the other tosses. The fact that you have gotten 8 tails in a row does NOT mean that you are "due" for a head. Before you start tossing the coin, however, the probability of getting 8 tails in a row is VERY small. It won't happen very often. Once it HAS happened, though, the probability of a ninth tail is still 0.5.
Same thing goes for baseball. Baseball?? Have you ever heard the announcer say that the batter has struck out 10 times in a row and is "due" for a hit? Same fallacy. Each atbat is an independent trial; previous attempts have no influence on the current one. The probability of a hit is just the same as before. If this batter is striking out a lot, maybe the batting coach had better get busy!
Remember: Coins and dice have no memory. If you flip a coin ten times and it happened to
come up heads all ten times, what is the probability that the eleventh flip will result in heads?
50%, the same as ONE trial. The gambler's fallacy or The Law of Averages
thinking is the mistaken belief that a tails result is "due".
mathmistakes.info is VERY useful.
The Texas Lottery keeps a record of the numbers that have won. WHY?!?!?
http://www.txlottery.org/export/sites/default/Games/Lotto_Texas/Number_Frequency.html
 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
 (a)
The answer is: all except (b). Even though there are apparent "hot streaks" of the same number in (a) and (ce), 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.
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
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 xNotice the peak at x.
A Visual Illustration: The 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.
Outline