Question

If a four-sided die is rolled and then a six-sided die is rolled, what is the probability that they will both show the number $2$?

• A. $\frac{1}{4}$
  
• B. $\frac{1}{6}$
  
• C. $\frac{1}{24}$
  
• D. $\frac{2}{3}$
  

Answer 1

The correct option is C $\frac{1}{24}$


The probability $\text{P(2)}$ of obtaining the number $2$ on the first die is:

Number of favorable outcomes $=$ Number of times number $2$ obtained $=1$
Total number of outcomes $=$ Total number of numbers on the die $=4$

$\text{P(2)}=\frac{\text{Number of times number 2 obtained}}{\text{Total number of numbers on the die}}$

$\text{P(2)}=\frac{1}{4}$

The probability $\text{P(2)}$ of obtaining the number $2$ on the second die is:

Number of favorable outcomes $=$ Number of times number $2$ obtained $=1$
Total number of outcomes $=$ Total number of numbers on the die $=6$

$\text{P(2)}=\frac{\text{Number of times number 2 obtained}}{\text{Total number of numbers on the die}}$

$\text{P(2)}=\frac{1}{6}$

As these events are independent events, the probability $\text{P(2,2)}$ of obtaining $2$ on both the dies is:

$\text{P(2,2)}=\frac{1}{4}\times \frac{1}{6}$

$\text{P(2,2)}=\frac{1}{24}$


The probability $\text{P(2,2)}$ of obtaining $2$ on both the dies is $\frac{1}{24}$.



$\underset{–}{AlternateSolution:}$

The sample space $\text{S}$ obtained when two dies, one of four-sided and the other of six-sided is rolled is given as:

$\text{S}=\left\{\right(1,1),(1,2),(1,3),(1,4),(1,5),$
$(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),$
$(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),$
$(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),$
$(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),$
$(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),$
$(6,6)\}$

Favorable outcome $=\left\{\right(2,2\left)\right\}$
Number of favorable outcome $=1$
Total number of outcomes $=24$

$\text{P(2,2)}=\frac{\text{Number of favorable outcome}}{\text{Total number of outcomes}}$

$\text{P(2,2)}=\frac{1}{24}$

The probability $\text{P(2,2)}$ of obtaining $2$ on both the die is $\frac{1}{24}$.

Therefore, option (c.) is the correct answer.