Question

Dialling a telephone number to his daughter an old man forgets the last two digits and dialled at random remembering only that the last two digits are different. The probability that the number dialled is correct is:
(a) \[\dfrac{1}{10}\]
(b) \[\dfrac{1}{45}\]
(c) \[\dfrac{1}{90}\]
(d) \[\dfrac{1}{135}\]

Answer 1

Hint: We solve this problem by considering the possible outcomes of the required number and the total number of outcomes.
We consider that there are two places where the digits 0 to 9 can be placed so as to get the 10 digit number the correct number of his daughter.
We have the formula of probability given as
\[P=\dfrac{\text{number of possible outcomes}}{\text{total number of outcomes}}\]

Complete step by step answer:
We are given that the old man forgot the last two digits of the telephone number.
We know that there are 10 digits from 0 to 9 which can be placed in the last places.
Let us assume that there are two boxes representing the last two digits of the telephone number as follows

Let us take the number of possibilities for the first box.
We know that the first box can take any digit from 0 to 9 which give 10 possibilities
Now, let us write the number of possibilities in the first box then we get

We are given that the digits are different.
Here, we can see that we placed one digit in the first box so that there will be 9 possibilities for the second box.
Now, by writing the number of possibilities in the second box we get

Let us assume that the total number of outcomes as \[N\]
Here, we can see that the total number of ways is the permutations of possibilities of each box.
By using the above condition we get the total number of outcomes as
\[\begin{align}
  & \Rightarrow N=10\times 9 \\
 & \Rightarrow N=90 \\
\end{align}\]
Let us assume that the number of possible outcomes of getting correct number as \[n\]
We know that there will be only one correct number for his daughter
By using the above condition we get the number of possible outcomes as
\[\Rightarrow n=1\]
We know that the formula of probability given as
\[P=\dfrac{\text{number of possible outcomes}}{\text{total number of outcomes}}\]
By using the above formula we get the probability of getting correct number as
\[\begin{align}
  & \Rightarrow P=\dfrac{n}{N} \\
 & \Rightarrow P=\dfrac{1}{90} \\
\end{align}\]
Therefore, we can conclude that the required probability as \[\dfrac{1}{90}\]
So, option (c) is the correct answer.

Note:
We can find the total number of outcomes by using the permutations concept.
Let us assume that the total number of outcomes as \[N\]
We have the number of ways of arranging the \[n\] objects in \[r\] places is given as \[{}^{n}{{P}_{r}}\] where
\[{}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}\]
By using the above condition we get the total number of outcomes by arranging the 10 digits in 2 places as
\[\begin{align}
  & \Rightarrow N={}^{10}{{P}_{2}} \\
 & \Rightarrow N=\dfrac{10!}{\left( 10-2 \right)!} \\
 & \Rightarrow N=\dfrac{10\times 9\times 8!}{8!}=90 \\
\end{align}\]
Let us assume that the number of possible outcomes of getting correct number as \[n\]
We know that there will be only one correct number for his daughter
By using the above condition we get the number of possible outcomes as
\[\Rightarrow n=1\]
We know that the formula of probability given as
\[P=\dfrac{\text{number of possible outcomes}}{\text{total number of outcomes}}\]
By using the above formula we get the probability of getting correct number as
\[\begin{align}
  & \Rightarrow P=\dfrac{n}{N} \\
 & \Rightarrow P=\dfrac{1}{90} \\
\end{align}\]
Therefore, we can conclude that the required probability as \[\dfrac{1}{90}\]
So, option (c) is the correct answer.