91Ó°ÊÓ

Tr denote the number of flips required to obtain a run of r consecutive heads

$p=$the probability that a coin lands on heads

$E\left[T_r\right]=$the number of flips required to obtain a run of r consecutive heads.

Find $E\left[T_râˆ\pounds T_{r−1}\right]$

$E\left[T_râˆ\pounds T_{r−1}\right]=T_{r−1}+1+(1−p)E\left[T_r\right]$

It has been determined that$E\left[T_râˆ\pounds T_{r−1}\right]=T_{r−1}+1+(1−p)E\left[T_r\right]$

Tr denote the number of flips required to obtain a run of r consecutive heads

Find $E\left[T_r\right]$

If expectations on both sides of (a) yields,

$E\left[T_r\right]=E\left[T_{r−1}\right]+1+(1−p)E\left[T_r\right]$

$=\frac{1}{p}+\frac{1}{p}E\left[T_{r−1}\right]$

It has been determined that$E\left[T_r\right]=\frac{1}{p}+\frac{1}{p}E\left[T_{r−1}\right]$̣

Tr denote the number of flips required to obtain a run of r consecutive heads

Find $E\left[T_1\right]$

Substitute 1 for r in part (b).

$E\left[T_1\right]=E\left[T_{1−1}\right]+1+(1−p)E\left[T_1\right]$

$=E\left[T_0\right]+1+(1−p)E\left[T_1\right]$

$=\frac{1}{p}+\frac{1}{p}E\left[T_0\right]$

$=\frac{1}{p}$

$\left[E\left[T_0\right]=0\right]$

It has been found that$E\left[T_1\right]=\frac{1}{p}$

Tr denote the number of flips required to obtain a run of r consecutive heads

Find $E\left[T_r\right]$

$E\left[T_r\right]=\frac{1}{p}+\frac{1}{p}E\left[T_{r−1}\right]$

$=\frac{1}{p}+\frac{1}{p}\left[\frac{1}{p}+\frac{1}{p}E\left[T_{r−1}\right]\right]$

$=\left(\frac{1}{p}\right)+{\left(\frac{1}{p}\right)}^2+{\left(\frac{1}{p}\right)}^{2}E\left[T_{r−2}\right]$

$=\left(\frac{1}{p}\right)+{\left(\frac{1}{p}\right)}^2+{\left(\frac{1}{p}\right)}^3+{\left(\frac{1}{p}\right)}^{3}E\left[T_{r−3}\right]$

$=\underset{i=1}{\overset{r}{∑}} {\left(\frac{1}{p}\right)}^i+{\left(\frac{1}{p}\right)}^{r}E\left[T_0\right]$

$=\underset{i=1}{\overset{r}{∑}} {\left(\frac{1}{p}\right)}^i$

It has been found that$E\left[T_r\right]=\underset{i=1}{\overset{r}{∑}} {\left(\frac{1}{p}\right)}^i$