Question

A and B throw with one dice for a stake of Rs. $99$ which is to be won by the player who first throws $4$. If A has the first throw then their expectations of rupees are respectively -

• A. $50$ and $49$
• B. $54$ and $45$
• C. $45$ and $54$
• D. none

Answer 1

The correct option is B $54$ and $45$

Probability of A winning

$=\frac{1}{6}+\frac{5}{6}.\frac{5}{6}.\frac{1}{6}+\frac{5}{6}.\frac{5}{6}.\frac{5}{6}.\frac{5}{6}.\frac{1}{6}+.......$

$=\frac{1}{6}(1+{\left(\frac{5}{6}\right)}^2+{\left(\frac{5}{6}\right)}^4+........)$

$=\frac{1}{6}\left(\frac{1}{1-\frac{25}{36}}\right)=\frac{36}{6\left(11\right)}$

$=\frac{6}{11}$

$\therefore$expectation of rupees of A $=\frac{6}{11}\times 99$

$=54Rs$

and expectation of ruppes of B $=99-54$$=45Rs$