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Flip the coin until a head appears.

Number of flips needed $=N$

$P\{Nâ‰\yen I\},iâ‰\yen 1=?$

We have,

$∴P[N=i]={∫}_0^1 \left(\begin{array}{l}i−1\\ 0\end{array}\right)p^{\mathrm{′}}(1−p{)}^{i−1}dp$

$\begin{array}{r}={∫}_0^1 \mathrm{p}(1−\mathrm{p}{)}^{\mathrm{i}−1}\mathrm{dp}\end{array}$

localid="1647430523751" $=\frac{\left(1\right)!(i−1)!}{(2+i−1)!}$

localid="1647429441069" $=\frac{(i−1)!}{(i+1)!}$

$=\frac{1}{i(i+1)}$

$\begin{array}{r}∴\mathrm{P}[\mathrm{N}â‰\yen \mathrm{i}]=\underset{x=i}{\overset{\mathrm{∞}}{∑}} \frac{1}{x(x+1)}\end{array}$

localid="1647429510678" $=\underset{x=i}{\overset{\mathrm{∞}}{∑}} \left[\frac{1}{x}−\frac{1}{x+1}\right]$

$=\frac{1}{i}$

Hence$\mathrm{P}[\mathrm{N}â‰\yen i]=\frac{1}{\mathrm{i}};i=0,1,2,….,n$

Hence, the value of$P\{Nâ‰\yen i\},iâ‰\yen 1\text{is}\mathrm{P}[\mathrm{N}â‰\yen i]=\frac{1}{i};i=0,1,2,…,n$

Flip the coin until a head appears.

Number of flips needed$=N$

$P\{N=i\}=?$

We have,

$\begin{array}{r}\mathrm{P}[\mathrm{N}=\mathrm{i}]=\mathrm{P}[\mathrm{N}â‰\yen i]−\mathrm{P}[\mathrm{N}>i]\end{array}$

$\begin{array}{r}=\mathrm{P}[\mathrm{N}â‰\yen i]−\mathrm{P}[\mathrm{N}â‰\yen i+1]\end{array}$

$=\frac{1}{i}−\frac{1}{i+1}$

localid="1647429927042" $=\frac{1}{i(i+1)}$

$∴\mathrm{P}[\mathrm{N}=\mathrm{i}]=\frac{1}{\left(i\right)(i+1)};i=0,1,2,….,n$

Hence, the value of $E\left[N\right]$is$∞.$

Flip the coin until a head appears.

Number of flips needed $=N$

The Value of$E\left[N\right]=?$

$\begin{array}{r}\mathrm{E}\left(\mathrm{N}\right)=\underset{i=0}{\overset{\mathrm{∞}}{∑}} \mathrm{P}[\mathrm{N}â‰\yen i]\end{array}$

role="math" localid="1647431080445" $=\underset{i=0}{\overset{\mathrm{∞}}{∑}} \left(\frac{1}{i}\right)$

$=\mathrm{∞}$

Therefore, the value of$E\left[N\right]=∞$