Monday, February 28, 2005

Math Problem IV

There are approximately 6.3 billion (6,300,000,000) human beings on the Earth at the present moment.

Imagine that each person on the planet is given a coin which is scrupulously calibrated to be absolutely fair, equally likely to come up heads or tails. (Persons who are neurologically undeveloped, damaged, or ill will have a coin flipped on their behalf by a family member or neighbor.) At exactly midnight, GMT, everyone flips his or her coin and records the outcome (heads or tails). Then the process is repeated again at 12:02 AM, and 12:04 AM, and so forth, every two minutes, until every coin has come up heads at least once and tails at least once, at which point the coin-flipping stops.

What time is it (GMT) when the coin-flipping stops?

Answer in comments.

Activity: How many times will the last coin-flipper have seen his/r coin come up either all heads or all tails? Is s/he likely to believe that the coin is fair and unweighted? Divide into small groups and discuss the emotional impact of this exercise on the coin-flipper if A) s/he is inclined to be skeptical of the claims of others or B) s/he is inclined to assign religious meaning to the exercise.

How would you defend the fairness of the coin to the skeptic in A)?

How would you explain to the believer in B) that s/he had not been singled out for special treatment by a deity, that inevitably a long string of H or T outcomes was going to happen to somebody?

4 comments:

Jessi Guilford said...

The time cannot be known with any certainty, since random coin-flips are not required to conform precisely to a 50-50 distribution. However, we can get an idea of when the coin-flipping would end by comparing successive powers of 2 to the population in question.

At 12:02 AM, 50% of the population is eliminated (people who got HT or TH). At 12:04 AM, another 25% of the original set is eliminated. At 12:06 AM, another 12.5% of the original set is eliminated. And so on:

12:00 AM - 6,300,000,000 participants
12:02 AM - 3,150,000,000 participants
12:04 AM - 1,575,000,000 participants
12:06 AM - 787,500,000 participants
12:08 AM - 393,750,000 participants
12:10 AM - 196,875,000 participants
12:12 AM - 98,437,500 participants
12:14 AM - 49,218,750 participants
12:16 AM - 24,609,375 participants
12:18 AM - 12,304,688 participants
12:20 AM - 6,152,344 participants
12:22 AM - 3,076,172 participants
12:24 AM - 1,538,086 participants
12:26 AM - 769,043 participants
12:28 AM - 384,521 participants
12:30 AM - 192,261 participants
12:32 AM - 96,130 participants
12:34 AM - 48,065 participants
12:36 AM - 24,033 participants
12:38 AM - 12,016 participants
12:40 AM - 6,008 participants
12:42 AM - 3,004 participants
12:44 AM - 1,502 participants
12:46 AM - 751 participants
12:48 AM - 376 participants
12:50 AM - 188 participants
12:52 AM - 94 participants
12:54 AM - 47 participants
12:56 AM - 23 participants
12:58 AM - 12 participants
1:00 AM - 6 participants
1:02 AM - 3 participants
1:04 AM - 1 participant
1:06 AM - end

We can therefore be relatively certain that the coin-flipping would still be going on an hour later for somebody.

Anonymous said...

But it's entirely possible that someone might still be flipping their coin 24 or so hours later, right? How unlikely is that?

Jessi Guilford said...
This comment has been removed by a blog administrator.
Jessi Guilford said...

If someone were still flipping their coin 24 hours later, then divine intervention or a weighted coin becomes a possibility, though it's more likely that the coin-flipper in question has decided to ignore the rules and has developed some kind of obsessive-compulsive coin-flipping behavior. Here's why:

24 hours = 720 coin-flip cycles. Since the odds of heads or tails is 50%, the odds of anybody's flips coming up all heads, or all tails, is (.5)^720, which is roughly equivalent to a chance of 1 in 5.5x10^216 (5,500,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000).

This means, among other things, that if you had more than one planet with an Earth-sized population, and they were all participating in the coin-flip experiment, you would need more planets (8.755x10^206 planets) than there are atoms in the universe for it to be likely that one of the planets had someone still flipping coins (by the rules) 24 hours later.

It's not, literally, impossible. But it may as well be, at least for this planet, in this universe.

One cannot really assign probability numbers to things like the odds of a biased coin, but whatever the odds are, they are almost certain to be better than 1 in 5.5x10^216.

In fact, I'd say the combined odds that 1) there is a God, and 2) S/He cares about the outcome of the coin-flipping, and 3) S/He takes active steps to influence said outcome, look better than the odds of an honest flipper still going at it the following day.