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 at-bat 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?!?!?

  • 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.

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.

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.

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.