In a class of 35, students are numbered from 1 to 35. The ratio of boys to girls is 4 : 3. The roll numbers of students begin with boys and end with girls. Find the probability that a student selected is either a boy with prime roll number or a girl with composite roll number or an even roll number.

#### Solution

Sample space (S) = {1, 2, 3, …, 35}

n(S) = 35

Total number of students = 35

Number of boys = `4/7 xx 35`

= 20 ...[Boys Numbers = {1, 2, 3, …, 20}]

Number of girls = `3/7 xx 35`

= 15 ...[Girls Numbers = {21, 22, …, 35}]

Let A be the event of getting a boy role number with prime number

A = {2, 3, 5, 7, 11, 13, 17, 19}

n(A) = 8

P(A) = `("n"("A"))/("n"("S")) = 8/35`

Let B be the event of getting girls roll number with composite number.

B = {21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35}

n(B) = 12

P(B) = `("n"("B"))/("n"("S")) = 12/35`

Let C be the event of getting an even roll number.

C = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34}

n(C) = 17

P(C) = `("n"("C"))/("n"("S")) = 17/35`

n(A ∩ B) = 0

⇒ P(A ∩ B) = 0

(B ∩ C) = {22, 24, 26, 28, 30, 32, 34}

n(B ∩ C) = 7

P(B ∩ C) = `("n"("B" ∩ "C"))/("n"("S")) = 7/35`

(A ∩ C) = {2}

n(A ∩ C) = 1

P(A ∩ C) = `("n"("A" ∩ "C"))/("n"("S")) = 1/35`

A ∩ B ∩ C = { }

n(A ∩ B ∩ C) = 0

P(A ∩ B ∩ C) = 0

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(B ∩ C) − P(A ∩ C) + P(A ∩ B ∩ C)

= `8/35 + 12/35 + 17/35 - 0 - 7/35 - 1/35 + 0`

= `8/35 + 12/35 + 17/35 - 8/35`

= `(8 + 12 + 17 - 8)/35`

= `29/35`

Probability of getting roll number is `29/35`