Incoming students to a certain school take a mathematics placement exam. The pos
ID: 3398179 • Letter: I
Question
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.Explanation / Answer
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]