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Are the odds of flipping a coin and getting 20 heads without getting 2 tails in a row the same as flipping a coin and getting 10 heads without getting 1 tail?

Caleb Stanford
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Triumph
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  • Do you want the odds, or the probability? They are different and the answers so far have addressed probability, not odds. –  Jul 31 '16 at 00:49
  • @tilper As far as my understanding of what "odds" are, it doesn't matter either way. If not, then what's your definition of "odds"? – Caleb Stanford Jul 31 '16 at 01:25
  • Yeah in this case it doesn't matter. Didn't realize until after I'd closed the browser. Meh. –  Jul 31 '16 at 01:26
  • Please use titles that are as specific and descriptive as possible! – Caleb Stanford Jul 31 '16 at 01:28
  • Please take note of the comments under the answer you accepted. If this is indeed the answer to the question you intended to ask, please edit the question accordingly. – joriki Jul 31 '16 at 04:38
  • (The author's comment I was mainly referring to has been converted to an edit in the answer in the meantime.) – joriki Jul 31 '16 at 05:10

1 Answers1

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For the second experiment, the probability is simply $\left(\frac12\right)^{10}$.

For the first experiment, divide it into $20$ experiments where you flip a coin, then if it lands tails you flip it again, and the individual experiments succeed unless they yield tails twice. Each successful experiment yields heads once. Thus the overall experiment succeeds exactly if all $20$ experiments succeed, and the success probability for each of them is $\frac34$, so the overall success probability is

$$\left(\frac34\right)^{20}=\left(\frac9{16}\right)^{10}\gt\left(\frac12\right)^{10}\;.$$

joriki
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  • suppose we get head in first toss and tail in second toss , tail in third toss, head in fourth toss. (i.e., HTTH). Here, first experiment is success(as it it HT) and second experiment is also success (TH) but, two tails happened in row (second and third) and so overall experiment failed. Am I wrong here? – Kiran Jul 30 '16 at 20:48
  • @Kiran: Yes, you are. The first experiment ends after the first toss of heads, since you only flip again if the result was tails. The second experiment consists of two tails and thus fails. – joriki Jul 30 '16 at 20:55
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    @msinghal No, you are wrong. The experiments are independent. The denominator is not a factor of $2^{20}$. The problem as stated by the OP is flip until you get 20 heads, not flip 20 coins. – Caleb Stanford Jul 31 '16 at 04:30
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    Great answer @joriki. I admit I'm baffled so many misread such a clear question and upvoted/downvoted accordingly. – Caleb Stanford Jul 31 '16 at 04:31
  • @6005: Me too :-) I don't think I've ever seen this happen to such an extreme extent. – joriki Jul 31 '16 at 04:34
  • Ah, yes. I see I misread the question. I apologize for any confusion I caused; this answer is correct. – msinghal Jul 31 '16 at 04:59