Question

Consider the following experiment :
$\text{Step 1.}$ Flip a fair coin twice.
$\text{Step 2.}$ If the outcomes are (TAILS, HEADS), then output $Y$ and stop.
$\text{Step 3.}$ If the outcomes are either (HEADS, HEADS) or (HEADS, TAILS), then output $N$ and stop.
$\text{Step 4.}$ If the outcoms are (TAILS, TAILS), then go to $\text{Step 1.}$.

The probability that the output of the experiment is $Y$ is $\frac{1}{k}$. Then $k$ is

Answer 1

From the tree diagram,
$P\left(\text{output=Y}\right)=0.5\times 0.5+0.5\times 0.5\times 0.5\times 0.5+\dots$
It's an infinite G.P. with
$a=0.5\times 0.5$
$r=0.5\times 0.5$
$\therefore P\left(\text{output=Y}\right)=\frac{0.25}{1-0.25}=\frac{1}{3}$
Hence $k=3$