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

Hint: 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.