Question: A Book contains 1000 pages numbered consecutively. The probability that the sum of the digits of the...

Question

A Book contains 1000 pages numbered consecutively. The probability that the sum of the digits of the marked number of a page is 9 is
A. $\dfrac{55}{1000}$
B. $0$
C. $\,\dfrac{33}{1000}$
D. $\,\dfrac{44}{1000}\,$

Answer 1

Solution

According to the question book contains 1000 pages that means the total number of probability will be 1000. Sum of digit is 9 that means if there 1 digit will $\,1$ such number and for 2 digit will be $\,9$ such numbers for and similarly for 3 digit will be $\,45$ .Therefore, total number of outcomes is $\,55$ .

Complete step by step answer:
Total number of pages in the book contains $=1000$ .
Sum of digit $=9$
Favorable outcome for 1 digit is given by $\,\,9$ .
That is there is only $\,1$ such a number for 1 digit.
Favorable outcome for 2 digit is given by $9$ such number
That means possible outcome for 2 digit $=18,27,36,45,54,56,72,81,90$
That is, a favorable outcome of 2 digits will be $9$ such numbers.
Similarly, Favorable outcome for 3 digits will be 45.
That means,
There are $\,9$ such numbers
$108,117,126,\,....................\,\,.180\,$
There are $\,8$ such numbers
$207,216,225,\,......................\,.\,\,270$
There are $\,\,7$ such numbers
$306,315,\,.................................360$
There are $\,\,6$ such numbers
$405,414,\,.................................450$
Similarly, further we can solve this we get:
$-$
$-$
Similarly,
There are $\,\,2$ such numbers
$801,810$
There are $\,\,\,1$ such numbers
$900$
Total such numbers for 3 digit $=9+8+7+6+5+4+3+2+1$
Total such numbers for 3 digit $=45$
Hence, the total number of favorable outcome $=1+9+45$
The total number of favorable outcome $=10+45$
The total number of favorable outcome $=55$
Therefore, the required probability $=\dfrac{55}{1000}$

So, the correct answer is “Option A”.

Note: Above solution can be preferred by solving such types of problems. Remember that there are a total number of pages in a book that is 1000, that is the total number of probability which can divide with possible outcomes. In this way we can solve problems in a similar manner.