Page opened from a book having a doublet as the page number

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Problem 1

A page from a book containing 100 pages is opened at random. Find the probability that a doublet is found on the page.

(Or)

A book has 100 pages. If a page is opened at random, find the probability for the page to have a two digit number with the same digits
Ans :
9
100

Solution

Total number of numbers

⇒ Number of pages in the book

= 100

Experiment :

Opening a page at random, from a book with 100 pages

Total Number of Possible Choices

= Number of ways in which a page can be opened out of the total 100 pages

⇒ n = 100C1
=
100
1
= 100

Let

A : the event of opening a page with a number that has a boublet

For Event A

Number of pages that have a doublet page number within 100

= 9

{(11), (22), (33), (44), (55), (66), (77), (88), (99)}

Favorable
(doublets) 		Unfavorable
(Others) 		Total
Available 9 91 100
To Choose 1 0 1
Choices 9C191C0100C1

Number of Favorable Choices

= Number of ways in which one page with a page number which is a doublet can be opened from the total 9 favorable pages

⇒ mA = 9C1
=
9
1
= 9

Probability of opening a page with a page number that is a doublet

⇒ Probability of occurrence of Event A

=
Number of Favorable Choices for the Event
Total Number of Possible Choices for the Experiment
⇒ P(A) =
mA
n
=
9
100

Odds

Number of Unfavorable Choices

= Total Number of possible choices − Number of Favorable choices

⇒ mAc = n − mA
= 100 − 9
= 91

in favor

Odds in Favor of opening a page with a number that is a doublet

⇒ Odds in Favor of Event A

= Number of Favorable Choices : Number of Unfavorable Choices

= mA : mAc

= 9 : 91

against

Odds against opening a page with a number that is a doublet

⇒ Odds against Event A

= Number of Unfavorable Choices : Number of Favorable Choices

= mAc : mA

= 91 : 9

Author : The Edifier
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