Question

A card is drawn and replaced in an ordinary pack of playing cards. The number of times a card must be drawn so that the probability of getting atleast a club card is greater than $\frac{3}{4}$ is

• A. $7$
• B. $6$
• C. $5$
• D. $4$

Answer 1

Answer: (C)

$5$

Here the probability of success is
$\frac{13}{52}$ ...( there are $13$ club cards in a deck of $52$ cards)
$=\frac{1}{4}$
Hence $p=\frac{1}{4}$

The probability of getting at least a club card in $n$ no. of trials is
$1-{(1-\frac{1}{4})}^n>\frac{3}{4}$
$\Rightarrow 1-{\left(\frac{3}{4}\right)}^n>\frac{3}{4}$
$\Rightarrow \frac{1}{4}>{\left(\frac{3}{4}\right)}^n$

Taking $\log$ on both the sides, we get
$n>\frac{-\ln \left(4\right)}{\ln \left(3\right)-\ln \left(4\right)}$
$\Rightarrow n>4.81$

However, $n$ has to be a natural number.
Hence $n=5$