Number chosen from a subset of the set of natural numbers following a certain rule

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Problem 8

A number x is chosen at random out of the natural numbers 1, 2, 3,... 100. Find the probability for x to follow x +
100
x
> 50
Ans :
14
25

Solution

Experiment :

Choosing a number from the set of natural numbers from 1, 2, 3, ..., 100

Total Number of Possible Choices

= Number of ways in which a number can be chosen from the natural numbers from 1 to 100

⇒ n = 100C1
=
100
1
= 100
A : the event of the number x chosen following
100
x
> 50

For Event A

100
x
> 50
⇒
x2 + 100
x
> 50

⇒ x2 +100 > 50 x

⇒ x2 − 50x + 100 > 0

⇒ 2 ≥ x ≥ 47

⇒ x = {1, 2, 47, 48, 49, ... , 100}

Number of numbers that follow the rule

= 56

{1, 2, 47, 48, 49, ... , 100}

Favorable
(numbers following the rule) 		Unfavorable
(Others) 		Total
Available 56 44 100
To Choose 1 0 1
Choices 56C144C0100C1

Number of Favorable Choices

= Number of ways in which a number that follows the rule can be chosen from the total 56 favorable numbers

⇒ mA = 56C1
=
56
1
= 56

Probability of choosing a number which follows the rule

⇒ 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
=
56
100
=
14
25

Odds

Number of Unfavorable Choices

= Total Number of possible choices − Number of Favorable choices

⇒ mAc = n − mA
= 56 − 100
= 44

in favor

Odds in Favor of choosing a number that follows the rule

⇒ Odds in Favor of Event A

= Number of Favorable Choices : Number of Unfavorable Choices

= mA : mAc

= 56 : 44

= 14 : 11

against

Odds against choosing a number that follows the rule

⇒ Odds against Event A

= Number of Unfavorable Choices : Number of Favorable Choices

= mAc : mA

= 44 : 56

= 11 : 14

Author : The Edifier
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