Drawing an integer which is a multiple of two integers 6 and 8 having common multiples
Problem 6
| 2 |
| 25 |
Solution
Total number of integers
= 100
Experiment :
Choosing an integer from the first hundred positive integers
Total Number of Possible Choices
= Number of ways in which an integer can be chosen from the first hundred positive integers
| ⇒ n | = | 100C1 | ||
| = |
| |||
| = | 100 |
Let
A : the event of the integer chosen being a multiple of 6 and 8.
For Event A
From 1 to 100
Multiples of 6 ⇒ 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96
Multiples of 8 ⇒ 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96
24 which is the LCM of 6 and 8 and multiples of 24 appear in both the multiple's list.
Multiples of 6 and 8
= 8
{12, 24, 36, 48, 60, 72, 84, 96}
| Favorable (Multiples of 6 and 8) | Unfavorable (Others) | Total | |
|---|---|---|---|
| Available | 8 | 92 | 100 |
| To Choose | 1 | 0 | 1 |
| Choices | 8C1 | 92C0 | 100C1 |
Number of Favorable Choices
= Number of ways in which an integer which is a multiple of 6 and 8 can be drawn from the total 24 favorable integers
| ⇒ mA | = | 8C1 | ||
| = |
| |||
| = | 8 |
Probability of choosing an integer which is a multiple of 6 and 8
⇒ Probability of occurrence of Event A
| = |
|
| ⇒ P(A) | = |
| ||
| = |
| |||
| = |
|
Odds
= Total Number of possible choices − Number of Favorable choices
| ⇒ mAc | = | n − mA |
| = | 100 − 8 | |
| = | 92 |
in favor
Odds in Favor of choosing an integer which is a multiple of 6 and 8⇒ Odds in Favor of Event A
= Number of Favorable Choices : Number of Unfavorable Choices
= mA : mAc
= 8 : 92
= 2 : 23
against
Odds against choosing an integer which is a multiple of 6 and 8⇒ Odds against Event A
= Number of Unfavorable Choices : Number of Favorable Choices
= mAc : mA
= 92 : 8
= 23 : 2