Incoming students to a certain school take a mathematics pla

Incoming students to a certain school take a mathematics placement exam. The possible scores are 1, 2, 3. and 4. From past experience, the school knows that if a particular student\'s score is x {1,2,3,4}, then the student will become a mathematics major with probability Suppose that the incoming class had the following scores: 10% of the students scored a 1, 20% scored a 2, 60% scored a 3, and 10% scored a 4. What is the probability that a randomly selected student from the incoming class will become a mathematics major? Express your answer as a fraction in lowest terms. Suppose a randomly selected student from the incoming class turns out to be a mathematics major. What is the probability that she scored a 4 on the placement exam? Express your answer as a fraction in lowest terms.

Solution

let A1 be the event that a randomly selected student scored 1

A2 be the event that a randomly selected student scored 2

A3 be the event that a randomly selected student scored 3

A4 be the event that a randomly selected student scored 4

M be the event that the randomly selected student will become a mathematics major.

so given that

P[A1]=0.10   P[A2]=0.20     P[A3]=0.60      P[A4]=0.10

and P[M|A1]=(1-1)/(1+3)=0   [putting x=1 in (x-1)/(x+3)]

P[M|A2]=(2-1)/(2+3)=1/5      [putting x=2 in (x-1)/(x+3)]

P[M|A3]=(3-1)/(3+3)=2/6=1/3      [putting x=3 in (x-1)/(x+3)]

P[M|A4]=(4-1)/(4+3)=3/7      [putting x=4 in (x-1)/(x+3)]

a) the probability that a randomly selected student from the incoming class will become a mathematics major is

P[M]=P[M|A1]*P[A1]+P[M|A2]*P[A2]+P[M|A3]*P[A3]+P[M|A4]*P[A4] [by theorem of total probability]

       =0*0.10+1/5*0.20+1/3*0.60+3/7*0.10

       =1/25+1/5+3/70=99/350 [answer]

b) suppose a randomly selected student from the incoming class turns out to be a mathematics major.

so the probability that she scored a 4 on the placement exam

P[A4|M]=P[M|A4]*P[A4]/P[M]   [by Bayes\' theorem]

            =3/7*0.10/(99/350)=(3/70)/(99/350)=5/33 [answer]