Question

A card is drawn and replaced in an ordinary pack of $52$ playing cards. Minimum number of times must a card be drawn so that there is at least an even chance of drawing a heart card is:
(A) $2$
(B) $3$
(C) $4$
(D) more than four

Answer 1

Hint: In the given question, we are given an ordinary pack of playing cards with $52$ playing cards in it. We have to find the minimum number of times a card must be drawn in order to have the least even chance of drawing a card of heart suit. The question involves the core concepts of probability. We will also be using logarithms and some of its properties in order to solve the inequality formed by interpreting the problem given to us.

Complete step by step solution:
We know that the probability of drawing a card of hearts suit is $\left({\displaystyle \frac{1}{4}}\right)$ .
So, the probability of not drawing a card of heart suit is $1-{\displaystyle \frac{1}{4}}=\left({\displaystyle \frac{3}{4}}\right)$ .
Now, we have to find the number of times must a card be drawn so that there is at least an even chance of drawing a heart card.
So, the probability of drawing a heart card must be greater than or equal to $\left({\displaystyle \frac{1}{2}}\right)$ .
 $\Rightarrow P\left(drawingaheart\right)⩾\left({\displaystyle \frac{1}{2}}\right)$
We can write $P\left(drawingaheart\right)$ as $\left[1-P\left(notdrawingaheart\right)\right]$ . So, we get,
 $\Rightarrow 1-P\left(notdrawingaheart\right)⩾\left({\displaystyle \frac{1}{2}}\right)$
 $\Rightarrow 1-P\left(notdrawingaheart\right)⩾\left({\displaystyle \frac{1}{2}}\right)$
Now, we know that the probability of not drawing a heart card in n attempts is ${\left({\displaystyle \frac{3}{4}}\right)}^n$ . So, we get,
 $\Rightarrow 1-{\left({\displaystyle \frac{3}{4}}\right)}^n⩾\left({\displaystyle \frac{1}{2}}\right)$
Shifting the terms in order to simplify the inequality, we get,
 $\Rightarrow 1-\left({\displaystyle \frac{1}{2}}\right)⩾{\left({\displaystyle \frac{3}{4}}\right)}^n$
 $\Rightarrow \left({\displaystyle \frac{1}{2}}\right)⩾{\left({\displaystyle \frac{3}{4}}\right)}^n$
Taking logarithm on both sides of the equation, we get,
 $\Rightarrow \log \left({\displaystyle \frac{1}{2}}\right)⩾\log {\left({\displaystyle \frac{3}{4}}\right)}^n$
Now, using the logarithm property $\log x^n=n\log x$ , we get,
 $\Rightarrow \log \left({\displaystyle \frac{1}{2}}\right)⩾n\log \left({\displaystyle \frac{3}{4}}\right)$
Isolating n in order to find the minimum value of n, we get,
 $\Rightarrow n⩽{\displaystyle \frac{\log \left({\displaystyle \frac{1}{2}}\right)}{\log \left({\displaystyle \frac{3}{4}}\right)}}$
Now, computing and simplifying the right side of the equation, we get,
 $\Rightarrow n⩽2.409$
So, the minimum number of times a card must be drawn so that there is at least an even chance of drawing a heart card is $2$ .
So, the correct answer is “Option A”.

Note: On should have basic knowledge of probability and variety of questions in the chapter before attempting such typical illustrations. One should take care while doing the calculations as it can be a tedious task sometimes. We should know the properties of logarithms as it helps us in solving some equations and inequalities easily.