Math 645: A Survey of Mathematical Problems I
Frank Sottile

• As I mentioned last week, statistics are subtle, and even probability is likely tricky. For that, I ask that you re-read the piece "Chance and Chanceability" again, as I have just done. It is a rather good introduction to probability, and should provide you with what you need to get started on the homework for this week. Inspired by the table on page 231 of "Chance and Chanceability", I flipped a coin (US Quarter) 100 times, in groups of 10 each. Knowing a bit about the physics of flipping coins, I let the coin fall to the floor (wood) so that it would bounce, hopefully randomizing the outcome. This is what I got.

  trials	10	20	30	40	50	60	70	80	90	100
  H T	5 5	4   6	6    4	4    6	4    6	5    5	2    8	6    4	8    2	4    6
  Cumulative	5 5	9 11	15 15	19 21	23 27	28 32	30 40	36 44	44 46	48 52

  Back to the physics of coin flipping, it has been shown (by Persi Diaconis) that if you place the coin in one hand, say heads up, and catch it in mid air, then it is more likely to land heads up. I would be interested sometime in running such an experiment.

  As for experiments, I am very sceptical of the result of Lazzerini who found such a good value for π.

• We will spend the next few weeks delving into graph theory. The reading assignment starts us in that direction.

• Dr. Geller's Lecture 3.
• Read Chapter VII of "Mathematics and the Imagination" on "Chance and Chanceability" again.
• Read the selection from "Dots and Lines" by Trudeau, number 6 in the course packet.
• Chapter 2 of A Survey of Mathematical Problems.

1. Make your own table of 100 coin flips, using one of the two paradigms: Coin landing (and bouncing) on a hard surface, or flipping with heads up and catching in hand.
2. Exercises 2.1, 2.2, 2.3, 2.4, and 2.5, as well as Problems 2.1 and 2.2. (For these, do indicate some of your work, especially the computations for 2.4.)