self doubt

Question

A fair coin is tossed until a head comes up for the first time. The probability of this happening on an odd number toss is?

Answer 1

Your Toss will look like

$\text{H,TTH,TTTTH,....}$

So probability will be

$\frac{1}{2}+\frac{1}{{2}^{3}}+\frac{1}{{2}^{5}}+...$

$\frac{1}{2} \times \frac{1}{1-\frac{1}{2^{2}}}=\frac{2}{3}$

Comments

Thank you

Answer 2

The probability of coming up heads on an odd toss is either:

(A) You come up heads on the first toss, or
(B) You come up tails on the first two tosses, then come up with heads on an odd toss

These two cases are mutually exclusive, so the total probability is the sum of the two cases

Call your final result P

$P=\frac{1}{2}+\frac{1}{2}.\frac{1}{2}P$

$\frac{3}{4}P=\frac{1}{2} $

$P=\frac{2}{3}$.