None of the two lottery tickets drawn from the tickets numbered 1 to 30 has a prime number

Problem 2

In a lottery with 30 tickets numbered 1 to 30, two tickets are drawn simultaneously. The probability that none of the two tickets drawn has a prime number is
Ans :
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
=
30 × 29
2 × 1
= 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 10C020C230C2

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
=
20 × 19
2 × 1
= 10 × 19
= 190

Probability that none of the tickets drawn has a prime number on it.

⇒ 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
=
190
435
=
38
87

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 −
38
87
=
87 − 38
57
=
49
87

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
87
:
49
87
= 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
87
:
38
87
= 49 : 38