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:
“Ezra Pound still lives in a village and his world is a kind of village and people keep explaining things when they live in a village.... I have come not to mind if certain people live in villages and some of my friends still appear to live in villages and a village can be cozy as well as intuitive but must one really keep perpetually explaining and elucidating?”
—Gertrude Stein (1874–1946)
“Crushed to earth and rising again is an author’s gymnastic. Once he fails to struggle to his feet and grab his pen, he will contemplate a fact he should never permit himself to face: that in all probability books have been written, are being written, will be written, better than anything he has done, is doing, or will do.”
—Fannie Hurst (1889–1968)
“What art thou that usurp’st this time of night,
Together with that fair and warlike form
In which the majesty of buried Denmark
Did sometimes march? By heaven I charge thee speak!”
—William Shakespeare (1564–1616)
“The moral equalizes all; enriches, empowers all. It is the coin which buys all, and which all find in their pocket. Under the whip of the driver, the slave shall feel his equality with saints and heroes.”
—Ralph Waldo Emerson (1803–1882)