None of the two lottery tickets drawn from the tickets numbered 1 to 30 has a prime number
Problem 2
38 |
87 |
Solution
Total number of tickets
= 30
Number of tickets drawn
= 2
Experiment :
Drawing 2 tickets from the 30 tickets
Total Number of Possible Choices
= Number of ways in which 2 tickets can be drawn from the 30 tickets
⇒ n | = | 30C2 | ||
= |
| |||
= | 15 × 29 | |||
= | 435 |
Let
A : the event of none of the tickets drawn having a prime number
For Event A
Prime Numbers :
= 10
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
Numbers which have exactly two distinct factors, 1 and itself.
Composite Numbers :
= 19
{4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30}
Numbers which have more than two distinct factors.
Others : Neither Composite nor Prime
= 1
{1}
Prime | Others | Total | |
---|---|---|---|
10 | 20 | 30 | |
To Choose | 0 | 2 | 2 |
Choices | 10C0 | 20C2 | 30C2 |
Number of Favorable Choices
= Number of ways in which 2 tickets with no prime number on it can be drawn from the total 30 tickets
= Number of ways in which 2 tickets can be drawn from the 20 others available
⇒ mA | = | 20C2 | ||
= |
| |||
= | 10 × 19 | |||
= | 190 |
Probability that none of the tickets drawn has a prime number on it.
⇒ Probability of occurrence of Event A
= |
|
⇒ P(A) | = |
| ||
= |
| |||
= |
|
Odds
Number of Unfavorable Choices= Total Number of possible choices − Number of Favorable choices
⇒ mAc | = | n − mA |
= | 435 − 190 | |
= | 245 |
in favor
Odds in Favor of none of the tickets drawn having a prime number on it.⇒ Odds in Favor of Event A
= Number of Favorable Choices : Number of Unfavorable Choices
= mA : mAc
= 190 : 245
= 38 : 49
against
Odds against none of the tickets drawn having a prime number on it⇒ Odds against Event A
= Number of Unfavorable Choices : Number of Favorable Choices
= mAc : mA
= 245 : 190
= 49 : 38
Odds (alternative)
Probability of non-occurrence of Event A
⇒ P(Ac) | = | 1 − P(A) | ||
= | 1 −
| |||
= |
| |||
= |
|
in favor
Odds in Favor of none of the tickets drawn having a prime number on it⇒ Odds in Favor of Event A
= | Probability of occurrence of the event : Probability of non-occurrence of the event | ||||
= | P(A) : P(Ac) | ||||
= |
| ||||
= | 38 : 49 |
against
Odds against none of the tickets drawn having a prime number on it⇒ Odds against Event A
= | Probability of non-occurrence of the event : Probability of occurrence of the event | ||||
= | P(Ac) : P(A) | ||||
= |
| ||||
= | 49 : 38 |