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Does the likelihood of an event increase with the number of times it does not occur?

I'm seriously thought about probability of pitching pennies.

Suppose that you threw the coin 10 times.

The results are below.

  1. $front$
  2. $back$
  3. $back$
  4. $front$
  5. $back$
  6. $front$
  7. $front$
  8. $front$
  9. $front$
  10. $front$

Now, you will throw the coin one more time.

Let's think about probability in this situation.

I thought that It will be more probability in "$back$" side than "$front$" side.

but, Mathematically, It still equal probability in both side.

What's the problem?

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I thought that It will be more probability in "back" side than "front" side. — Why did you think such a thing? The problem seems to be with your intuition. –  ShreevatsaR Jul 19 '11 at 18:27
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The difference is that common sense gets a lot of things wrong: en.wikipedia.org/wiki/Gambler%27s_fallacy Something like this which is actually true is en.wikipedia.org/wiki/Regression_toward_the_mean . –  Qiaochu Yuan Jul 19 '11 at 18:28
    
That was strange. I think I typed in the wrong question number. –  Qiaochu Yuan Jul 19 '11 at 18:42
    
Think about the problem the following way: I throw the coin 10 times, and don't tell you which is the outcome... Now I throw the coin the 11th time, which one is more probable? Tail or Heads? Keep in mind that my 10 throws could have been exactly your output, or maybe exactly the opposite.....Does the outcome of the 11th throw depend on you knowing the first 10 throws? –  N. S. Jul 19 '11 at 18:49
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marked as duplicate by Qiaochu Yuan Jul 19 '11 at 18:41

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4 Answers

up vote 1 down vote accepted

The odds of flipping a coin and it being front 6 times in a row is very low however this is not the bet you are making.

The bet you are making is what are the odds of flipping a front 6 times in a row given the fact that there were 5 previous fronts. That probability is 50/50. You can draw out the tree for 6 coin flips and you will realize that of the end nodes 50% are front and 50% are back.

Another way to say it is they are Independent Events

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As long as you're dealing with a fair coin, I think you've fallen for the Gambler's Fallacy. There's no reason to think that the coin is due for a certain outcome because it hasn't happened often enough.

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It is possible to encounter long sequences of heads or tails with a fair coin. You merely need to make more observations for these aberrations to even out. –  Emre Jul 19 '11 at 18:03
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Each coin flip is assumed independent of the rest and so the observed outcomes have no affect on the outcome of the next flip. The probability of a front occurring is $1/2$ and the probability of a back occurring is $1/2$. Anyway, why would you go against the evidence and assume that the rare event should happen? If you're actually making use of the empirical distribution then you should assume the next flip will be a front with high probability (0.7 in this case). –  Max Jul 19 '11 at 18:07
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At least I would interpret the statement

Empirically, It will be more probability in "back" side than "front" side.

as a statement about what I think is known as the empirical distribution associated to the sample you gave (intuitiviely this is simply the best guess you can make from just observing a sample), rather than the distribution you assume the sample to be taken from.

If interpreted like this, the statement that one outcome is "empirically" more probable than another simply means that you have observed that outcome more times, saying nothing about the distribution you assume the sample was taken from.

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The problem is in the Law of large numbers (wiki page) and due to the fact you coin is fair.

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@oleksii: Sorry but the law of large numbers has exactly nothing to say about the result of 11 throws of a penny. –  Did Jul 19 '11 at 22:21
    
@Didier, why do you disagree? If the coin is fair, then p(H) = p(T) = 0.5. This actually is not visible during the experiment of 11 throws. Such experiment can yield all 11 Heads and still the probability of the next throw will be 0.5. As the coin is fair and events are independent. Given the experiment conducted many times intuitive observations will approach the expected value. Could you please comment on my logic? –  oleksii Jul 20 '11 at 9:02
    
@oleksii You wrote the reason yourself: This actually is not visible during the experiment of 11 throws. As a consequence the sentence where you invoke many times is (true but) off topic. –  Did Jul 20 '11 at 12:16
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