A certain city has 4 school districts (West, North, Central, and Riverside). The

Question

A certain city has 4 school districts (West, North, Central, and Riverside). The following
table gives the breakdown, by level (elementary, middle, or high school) and district of
students in the city. A student is chosen at random. If the student is in high school, what
is the probability the student is in the Riverside district? If the student is in the Central
district, what is the probability the student is in elementary school? If the student is in
the North district, what is the probability that the student is in elementary school or
middle school? If the student is in elementary or middle school, what is the probability
that the student is in the North district? If the student is the West or North districts,
what is the probability that the student is not in middle school?
Elementary Middle High
West 484 255 444
North 492 274 505
Central 500 270 563
Riverside 405 202 398

Explanation / Answer

Use Bayes theorem to determine the probabilities.According to Bayes theorem, P(E1|B)=P(E1)*P(B|E1)/{P(E1)*P(B|E1)+P(E2)*P(B|E2)}

In the current context assume E1, E2, E3 and E4 denote event of selecting school from 4 districts and B1, B2 and B3 denote the event of selecting 3 type of schools.

Thus, P(E)=1/4, P(B)=1/3.

P(Riverside|High school)=P(Riverside)*P(High school|Riverside)/{P(Riverside)*P(High school|Riverside)+P(Central)*P(High school|Central)+P(North)*P(High school|North)+P(West)*P(High school|West)}

=1/4*0.396/(1/4*0.396+1/4*0.422+1/4*0.397+1/4*0.375) [note: P(High school|Riverside)=398/(398+405+202)]

=0.2491

P(Elementary school|Central district)=P(Elementary school)*P(Central district|Elementary school)/{P(Elemntary school)*P(Central district|Elementary school)+P(Middle school)*P(Central district|Middle school)+P(High)*P(Central District|High)}

=(1/3*0.21)/(1/3*0.21+1/3*0.27+1/3*0.29)

=0.27

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