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Does probability catch up with itself?


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For example, the probability of flipping heads with a two-faced coin is 1/2. Does the probability of flipping heads increase after flipping consecutive tails and vice versa?

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For example, the probability of flipping heads with a two-faced coin is 1/2. Does the probability of flipping heads increase after flipping consecutive tails and vice versa?

 

No, but casinos make a boatload of cash from people who think otherwise. There is a saying: dice have no memory. Applies to coins, too.

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No, but casinos make a boatload of cash from people who think otherwise. There is a saying: dice have no memory. Applies to coins, too.

So even though the probability of flipping heads 100 times consecutively is 1/2^100, you shouldn't necessarily EXPECT to flip tails after flipping 99 consecutive heads?

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For example, the probability of flipping heads with a two-faced coin is 1/2. Does the probability of flipping heads increase after flipping consecutive tails and vice versa?

 

No, those probabilities are independent.

 

So even though the probability of flipping heads 100 times consecutively is 1/2^100, you shouldn't necessarily EXPECT to flip tails after flipping 99 consecutive heads?

 

You should expect it 50% of the time, just like on the first flip.

 

However, the probabilities of 99 consecutive flips being identical are so low that it's far more likely that you actually have a two-headed coin.

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You might like to read the following link. There could be a theoretical answer in line with the theory "the coin has no memory". However, if you spun a coin (say) 10 times in a practical situation and it came up heads each time the coin may not be a "fair coin" and so you might have more than a 50/50 chance of getting another head on the next spin.

With a "fair coin" it is said that over a large number of spins the difference between the number of heads and the number of tails obtained will diminish. However, for all practical purposes all numbers of spin you use (even thousands) can be considered small numbers! With a fair coin the chance of getting a head when you spin is not affected by previous spins. It will be a 50/50 chance. http://en.wikipedia.org/wiki/Gambler%27s_fallacy

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If a coin came up heads 100 times in a row, I would bet on heads for the next flip. Seriously, 100 heads in a row is a strong indication that there is trick somewhere, like a two headed coin.

Right. But you cannot be 100% confident that it is an unfair coin.

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Right. But you cannot be 100% confident that it is an unfair coin.

 

You cannot be 100% confident of anything.

 

If someone flipped heads 99 times in a row, it's either an unfair coin or you are dreaming the entire episode.

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If someone flipped heads 99 times in a row, it's either an unfair coin or you are dreaming the entire episode.

 

..or it's one of those "one chance in 634 billion billion billion" events that make life interesting. :rolleyes:

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..or it's one of those "one chance in 634 billion billion billion" events that make life interesting. :rolleyes:

 

Seriously, those events only happen in fiction! If you see an event occur against those kinds of odds then it is far more likely that you have actually miscalculated the odds.

 

It reminds me of my old D&D days when we had a guy who insisted that he had rolled six 18s for one of his characters. Even when I pointed out that if every one of the 4.5 billion people on the planet rolled 100 characters, there will still be just a miniscule chance that even one of them would randomly roll six 18s on just one of their characters --- yet he still claimed it to be true.

 

For example, as a Risk player, I taught myself on a lark to roll 18s at will with dice just so I could flatten my friends in a game one night.

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One difficulty with probability theory is that you can never prove anything, except theorems about probability. Nevertheless you can draw inferences and you can become damn suspicious. It is a useful tool in the right hands. It can be badly misused by those who do not understand it.

 

Any event with any non-zero probability (like a string of 100 or even 1,000,000 consecutive heads in coin flips) will occur with probability 1 in an infinite number of trials. The operative word is "infinite" and not just "big".

 

Even probability zero events can occur, just not very often. So, it is possible that in an infinite number of trials with one particular coin, 100 consecutive heads will not occur. It is even possible that only tails will occur. In fact ANY string could occur. Moreover, the probability of any specific infinite sequence is 0, including whatever string is actually produced. This is simply a result of how probability measures on infinite product spaces are built.

 

So, when confronted by an event of extremely low probability you have two choices: 1) You can believe that the event occurred narurally as a result of fluctuations of the cosmos or 2) You can question the underlying assumptions. In my experience 2 is the most profitable course of action.

 

Example: Once I was looking at data from a small but critical component. The leakage current was low, "minus seven sigma". The guy responsible for the component thought this was just dandy since low leakage current is generally good, and "minus seven sigma" events still have a non-zero probability of occurrence. But minus seven sigma events do not occur very often, the world is not really normally distributed, and I thought it much more likely that something was amiss in the manufacturing process. There was a well-publicized and several billion $ project potentially depending on the component working properly. Over the strenuous objections of the other guy, the component was pulled and re-tested. It failed.

 

A lot depends on what is really known about the probability distribution, what is assumed, how many trials are involved and what is known about the associated physics. If you are gambling and a coin has shown 100 consecutive heads, I would suspect a skunk in the wood pile and expect either another head on the next toss or a deft switch of coin by the con man flipping it.

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