In this next part, we begin examining what sorts of outcomes we might
expect from a population that do not all have the same probability of
failure.
Suppose we have 512 true coins, and we flip each of them four times.
How many times do we expect the coin to land tails?
How many of the coins should we expect to land tails in all four trials?
The probability that a true coin lands tails is 1/2. We are flipping
512 coins 4 times each, so we ought to expect that half of those (1024) land tails.
Each flip of a true coin is independent of the previous outcomes, so the probability of a single coin landing tails four times in a row is
(1/2)^4 = 1/16. With 512 coins, we would expect 32 of them to land tails every time.
Let us repeat this experiment, except this time we shall used biased coins. We'll have coins that land tails 75% of the time, and coins that land tails 25% of the time; an equal number of each. Let us then ask the same two questions again.
How many times do we expect the coin to land tails?
We are flipping 256 biased coins 4 times each. So the coins that land tails 25% of the time should produce tails 256 times. The coins biased the other way should land tails 768 times. So we get the same answer as before.
How many of the coins should we expect to land tails in all four trials?
IF the probability of landing tails once is 1/4, then the probability of landing tails four times in succession is 1/256. If the probability is 3/4, then the probability of 4 in a row is 81/256. Since we have 256 coins of each type (a fortuitous coincidence, that), we expect 82 coins to land tails four times in a row.
In other words, even though the big picture looks like that of a population of true coins, the clumping of the runs of tails tells a truer story.
May 7, 2005 8:47 PM
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