Question

A biased coin with probability P ($0<P<1$) of heads, is tossed until a head appears for the first time. If the probability that the number of tosses required is even is $2/5$, then the value of P is?

• A. $1/4$
• B. $1/6$
• C. $1/3$
• D. $1/2$

Answer 1

The correct option is C $1/3$

Given the probability of heads = P

A head appears for $1^{st}$ time on second toss if the first

toss is tail and second is head

Probability for this $=(1-P)P$

($\because$ Either it is head or tail.Probability of getting tail = 1- Probability of getting head)

For the head to appear for first time on $4^{th}$

toss probability $=(1-P{)}^{3}P$

And Similarly for $6^{th},8^{th}$....tosses

Total Probability for getting head for first time

On even toss $=(1-P)P+(1-P{)}^{3}P+...(1-P{)}^{2n+1}P$

$=(1-P)P(1+(1+P{)}^2+(1-P{)}^4+....\infty )$

$=(1-P)P\left(\frac{1}{1-(1-P{)}^2}\right)$

$\Rightarrow (1-P)P\left(\frac{1}{(2-P)}\right)=\frac{2}{5}$

$\Rightarrow 5-5P=4-2P\Rightarrow P=\frac{1}{3}$

$\therefore P=\frac{1}{3}$