Question

Out of 50 tickets numbered 00,01,02,……..,49 one ticket is drawn randomly, the probability of the ticket having the product of its digits 7 given that the sum of the digits is 8, is
A. $ \dfrac{1}{14} $
B. $ \dfrac{3}{14} $
C. $ \dfrac{1}{5} $
D. None of these

Answer 1

Hint: To solve this question we will use the concept of probability. As we have to find the combined probability of the ticket having the product of its digits 7 given that the sum of the digits is 8, we will use the formula of combined probability which is given by $ P\left( A\bigcap B \right)=\dfrac{n\left( A\bigcap B \right)}{n(S)} $
Where, $ n\left( A\bigcap B \right) $ = number of possible events and
 $ n(S) $ = total number of sample space

Complete step by step answer:
We have been given that out of 50 tickets numbered 00,01,02,……..,49 one ticket is drawn randomly.
We have to find the probability of the ticket having the product of its digits 7 given that the sum of the digits is 8.
Now, let us consider $ n(S) $ = total number of sample space
We have 50 tickets so the value of $ n(S) $ will be $ n(S)=50 $
Now, let us consider A is the set of tickets having the product of its digits 7
So we get $ A=\left\{ 17 \right\} $
Let us consider B is the set of selected tickets having the sum of the digits 8
So we get $ B=\left\{ 08,17,26,35,44 \right\} $
Now, we have to find such a set which includes tickets having the product of its digits 7 and the sum of the digits 8.
So we get $ A\bigcap B=\left\{ 17 \right\} $
So we have \[n\left( A\bigcap B \right)=1\]
Now, the combined probability is given by $ P\left( A\bigcap B \right)=\dfrac{n\left( A\bigcap B \right)}{n(S)} $
Substituting the values in the above formula we get
  $ \Rightarrow P\left( A\bigcap B \right)=\dfrac{1}{50} $
The correct answer is not given in the options.
So option D- None of these is the correct answer.

Note:
When we have to consider the combined condition we always find the intersection of both the sets and then calculate the probability. Always remember the value of probability is not greater than 1. Avoid calculation mistakes and be careful while taking intersections. In intersection, we will choose the common terms from both sets.