Mathematics - Probability Question with Solution | TestHub

The sequence of consecutive heads may starts with $1^{\text{st }}$ toss or $2^{\text{nd }}$ toss or $3^{\text{rd }}$ toss ---- or at $6^{\text{th }}$ toss. In any case , if it starts with $r$ th throw , the first ( $r-2$ ) throws may be head or tail but $(r-1)$ st throw must be tail, after which again a head or tail can show up:

$$\text{ ∴ Probability }=\frac{1}{2^7}+\frac{1}{2}\cdot \frac{1}{2^7}+\frac{1}{2}\cdot \frac{1}{2^7}+\cdots +\frac{1}{2}\cdot \frac{1}{2^7}=\frac{1}{2^7}\left[1+\frac{5}{2}\right]=\frac{7}{2^8}$$

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