{No) Pasaran II

Answer the problem came with—cogent reason:

Flip the coin twice.

HH repeat two flips
HT grade is A
TH grade is B
TT repeat two flips

Comment by a friend to whom I gave P. I—insufficient reason:

This solution is unnecessary. On Day One, the students had no reason to expect that the coin—if unfair—would have either particular bias. Independently, they had equally no reason to expect either particular assignment of heads and tails. Therefore, on the day the students were required to decide whether or not to take the course for credit, the prospect of a simple single flip sufficed to guarantee the promised outcome.

(“Cogent reason” and “insufficient reason” refer to the two extreme philosophies for the assignment of probability. (But you knew that?))


  1. If the coin is so biased that it always come up heads or tails, the professor is going to be a while doing those grades!

  2. If the probability is 1.0 that the coin comes up heads, will it come up tails at least once after an infinite number of flips?

  3. There is no "after" an infinite number of flips.

    In any case, it depends if that 1.0 is exactly 1 (i.e. the coin has heads on both sides) or a rounding of, say, 0.99999999999987 .

  4. Oh, c'mon, guys...

  5. Okay.

    Chances of all heads forever: 1.0 ^ (inf) = 1.0 .

    Chances of a single tails: 1.0 - 1.0 = 0.0 .

  6. (sigh)

    OK, folks, leaving for a moment the metaphysical realm: What is the probability that--following one billion independent trials, each of which has a probability of success is one in a billion--not even one trial has succeeded?

  7. Oops--sorry about that sentence.

  8. 1-(1-1/1e9)^1e9 = 0.63, or 63%.

  9. Yes (No, I didn't check your arithmetic), i.e., for large numbers of trials, is 1/e.

  10. "...not even one trial"

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