Question

A four digit number is formed by the digits $1,2,3,4$ with no repetition. The probability that the number is odd is

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{1}{4}$
• D. $\frac{1}{6}$

Answer 1

The correct option is A

$\frac{1}{2}$

Explanation of the correct option:

Finding the required probability:

Let $S$ be the no. of ways in which a $4$ digit number can be formed using the given digit.

$\Rightarrow n\left(S\right)=4!$ (as repetition is not allowed, so thousands place can be filled in four ways, hundreds place in three ways, and so on.)

Let $A$ be the event that the number formed is an odd number.

So, the units place can be filled in $2$ ways. (Using $1$ or $3$)

The thousands place can be filled in $3$ ways. (by remaining three digits)

The thousands place can be filled in $2$ ways.

The tens place can be filled in $1$ way.

$\begin{array}{rcl}& \Rightarrow & n\left(A\right)=2\times 3\times 2\times 1\\ & =& 12\end{array}$

Therefore, the required probability is:

$\begin{array}{rcl}P\left(A\right)& =& \frac{n\left(A\right)}{n\left(S\right)}\\ & =& \frac{12}{4!}\\ & =& \frac{12}{4\times 3\times 2\times 1}\\ & =& \frac{1}{2}\end{array}$

Hence, the correct answer is option A.