Question

Probability of all $ 3 $ digits numbers having all the digits same is
 $ A)\dfrac{1}{{100}} $
 $ B)\dfrac{3}{{100}} $
 $ C)\dfrac{7}{{100}} $
 $ D) $ None of these

Answer 1

Hint: First, we need to know the concept of probability.
Probability is the term mathematically with events that occur, which is the number of favorable events that divides the total number of outcomes.
The total three-digit numbers can be found by the last digit subtracted by the starting digit of the three digits plus one.
After finding the total event, try to find the favorable event in which all the three digits are the same.
Formula used:
 $ P = \dfrac{F}{T} $ where P is the probability, F is the possible favorable events and T is the total outcomes from the given.
The total number of three-digit numbers is $ L - F + 1 $ where L is the last number in the three digits and F is the first number in the three digits.

Complete step by step answer:
Since we know that the three-digit numbers are from $ 100 $ to $ 999 $ (below $ 100 $ are two digits and after $ 999 $ are four digits)
Thus, the last three-digit number is $ 999 $ and the first three-digit number is $ 100 $
Hence applying the formula to get the total event as $ L - F + 1 = 999 - 100 + 1 \Rightarrow 900 $
Thus, we get the total outcomes from the given is $ 900 $ .
Now we are going to find the favorable events, which is all three digits are needs to be the same, like $ 111 $
So, in three digits the possible same three digits are $ 111,222,333,444,555,666,777,888,999 $ (all the three digits are exactly the same)
Hence, we get the favorable event as $ 9 $ possible outcomes. Now applying these values into the probability formula ( $ P = \dfrac{F}{T} $ where P is the probability, F is the possible favorable events and T is the total outcomes from the given)
Thus, we get $ P = \dfrac{F}{T} \Rightarrow \dfrac{9}{{900}} $ (where $ 9 $ possible outcomes are the favorable event and total outcomes from the given are $ 900 $ )
Solving the values, we get $ P = \dfrac{9}{{900}} \Rightarrow \dfrac{1}{{100}} $ (by division operation)

So, the correct answer is “Option A”.

Note: If we divide the probability value and multiply with the $ 100 $ then we get the actual percentage of the given value.
We are also able to solve without the formula for the Total number of three-digit numbers are $ L - F + 1 $ , because in the three-digit numbers trivially there are total $ 900 $ numbers are there.
 $ \dfrac{1}{{100}} $ means the total outcome is $ 100 $ and the possible way to get the same numbers is $ 1 $