Question

It is estimated that in a school 35% of the pupils play an instrument, 30% of the pupils attend the school Art club and 10% of the pupils do both, play an instrument and attend the school Art club. Every pupil chooses which activity to do independently of other pupils. We randomly choose 10 pupils from the school and record for each pupil whether they play an instrument and/or attend the school Art club.

(a) Describe an appropriate sample space.

(b) Compute the probability that among randomly chosen 10 pupils, exactly 4 pupils play an instrument and attend the school Art club, exactly 3 play an instrument but do not attend the school Art club and exactly 2 neither play an instrument nor attend the school Art club. Justify your answer.

(c) If among the randomly chosen 10 pupils at least 2 play an instrument compute the probability that at least 1 pupil does both activities.

Answer 1

Answer:

Given that:

It is estimated that in a school 35% of the pupils play an instrument, 30% of the pupils attend the school Art club and 10% of the pupils do both, play an instrument and attend the school Art club.

(a) Describe an appropriate sample space.

Let A be the event of playing an instrument

B be the event of attending school art club

P(A)= 0.35, P(B)= 0.30,P(A B) = 0.10

A randomly chosen student can either be a member of the event or   or

Sample space =

b) instrument and attend the school Art club, exactly 3 play an instrument but do not attend the school Art club and exactly 2 neither play an instrument nor attend the school Art club

Since there are more than two possibilities for this random experiment the required probability uses the multinomial probability low the probability distribution

where

According to the question n =10

c) If among the randomly chosen 10 pupils at least 2 play an instrument compute the probability that at least 1 pupil does both activities.

P(atleast 2 play an instrument)=1-p(none plays an instrument) -p(one plays an instrument)

Here consider playing an instrument on success and sent of events on failure.

Then p = 0.35 and q = 1-0.35=0.15

According to binomial probability law

required probability =

P(atleast one does both) =1-P(None does both)

Here consider both the happening on success i.e AB as success and its non happening i.e as failure.Again according to binomial probability law,

required probability

We know which implies atleast one does both c Atleast two play om instrument..(1)