Question

A person draws a card from a pack, replace it shuffles the pack, again draws a card, replace it and draws again. This he does until he draws a heart. The probability that he will have to make at least four draws is

• A. $\frac{27}{256}$
• B. $\frac{175}{256}$
• C. $\frac{27}{64}$
• D. None of these

Answer 1

Answer: (C)

$\frac{27}{64}$

Probability of drawing a heart $=\frac{13}{52}=\frac{1}{4}$
$P($ He required at least $4$ draws for heart$)$
$={\left(\frac{3}{4}\right)}^3.\frac{1}{4}+{\left(\frac{3}{4}\right)}^4.\frac{1}{4}+{\left(\frac{3}{4}\right)}^5.\frac{1}{4}+...$
$=\frac{{\left(\frac{3}{4}\right)}^3.\frac{1}{4}}{1-\frac{3}{4}}=\frac{27}{64}$