Explaining Why The Probability Is 1/2 For A Fair Coin
We can see from the above that, if one flips a fair coin 21 times, then the probability of 21 heads is 1 in 2,097,152. However, the probability of flipping a head after having already flipped 20 heads in a row is simply 1⁄2. This is an application of Bayes' theorem.
This can also be seen without knowing that 20 heads have occurred for certain (without applying of Bayes' theorem). Consider the following two probabilities, assuming a fair coin:
- probability of 20 heads, then 1 tail = 0.520 × 0.5 = 0.521
- probability of 20 heads, then 1 head = 0.520 × 0.5 = 0.521
The probability of getting 20 heads then 1 tail, and the probability of getting 20 heads then another head are both 1 in 2,097,152. Therefore, it is equally likely to flip 21 heads as it is to flip 20 heads and then 1 tail when flipping a fair coin 21 times. Furthermore, these two probabilities are equally as likely as any other 21-flip combinations that can be obtained (there are 2,097,152 total); all 21-flip combinations will have probabilities equal to 0.521, or 1 in 2,097,152. From these observations, there is no reason to assume at any point that a change of luck is warranted based on prior trials (flips), because every outcome observed will always have been as likely as the other outcomes that were not observed for that particular trial, given a fair coin. Therefore, just as Bayes' theorem shows, the result of each trial comes down to the base probability of the fair coin: 1⁄2.
Read more about this topic: Gambler's Fallacy
Famous quotes containing the words explaining, probability, fair and/or coin:
“Behold, Esau my brother is a hairy man, and I am a smooth man.”
—Bible: Hebrew Jacob, in Genesis, 27:11.
To his mother Rebekah, explaining how the blind Isaac might discover the ploy of his pretending to be Esau. “Esau was a cunning hunter, a man of the field; and Jacob was a plain man, dwelling in tents.” (25:27)
“Only in Britain could it be thought a defect to be “too clever by half.” The probability is that too many people are too stupid by three-quarters.”
—John Major (b. 1943)
“Let us have a fair field! This is all we ask, and we will be content with nothing less. The finger of evolution, which touches everything, is laid tenderly upon women. They have on their side all the elements of progress, and its spirit stirs within them. They are fighting, not for themselves alone, but for the future of humanity. Let them have a fair field!”
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“It is not funny that a man should be killed, but it is sometimes funny that he should be killed for so little, and that his death should be the coin of what we call civilization.”
—Raymond Chandler (1888–1959)