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Question:

A fair coin is tossed repeatedly until the sequence $HH$ of two consecutive heads appears, and the total number of tosses $n$ is recorded. Find $\Pr \left( {\left\{ {n = 2018} \right\}} \right)$? Hint given: Note that $n$ does not have a geometric/negative binomial distribution. Use the Law of Total Probability and condition on the outcome of the first two tosses and use the mutually exclusive and exhaustive events $\{HH\}$, $\{HT\}$, and $\{TH,TT\}$.

I know a similar question was asked here, but this question asks to apply the law of total probability. Hopefully I'm not wrong in thinking this would have a different approach.

If we suppose the first two tosses is $HH$. Then how do we apply the law of total probability to find $\Pr \left( n \right)$? Would it look like: $$\Pr \left( n \right) = \Pr \left( {n \mid HH} \right)\Pr \left( {HH} \right) + \Pr \left( {n \mid HT} \right)\Pr \left( {HT} \right) + \Pr \left( {n \mid TH} \right)\Pr \left( {TH} \right) + \Pr \left( {n \mid TT} \right)\Pr \left( {TT} \right)$$

If so, how does one find $\Pr \left( {n \mid HH} \right)$, $\Pr \left( {n \mid HT} \right)$, etc.?

pabhp
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    Well, I suppose the intent is to work recursively. Get an expression for $p_n$ in terms of $p_{n-1}, p_{n-2}$. – lulu Mar 28 '21 at 11:25
  • @lulu: What would $p_n$ represent? – pabhp Mar 28 '21 at 11:35
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    The probability that the first $HH$ is completed on trial $n$. Thus, the answer to your problem is $p_{2018}$. – lulu Mar 28 '21 at 11:36
  • @lulu: I'm still quite lost as to how to start off. How do I get the expression for $p_n$ in terms of $p_{n-1}$ and $p_{n-2}$? – pabhp Mar 28 '21 at 12:24
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    There are exactly two paths to victory in $n$ trials (for $n>2$): either you throw $T$ first and then achieve $HH$ in exactly $n-1$ trials, or you throw $HT$ first and then achieve $HH$ in exactly $n-2$ trials. – lulu Mar 28 '21 at 12:26
  • @lulu: How does the law of total probability come into play then? – pabhp Mar 28 '21 at 14:48
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    Well, the total probability is, of course, the sum of the probabilities along those two paths. – lulu Mar 28 '21 at 14:50

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