1000 candidates take SOA exam P. Countrywide, 35% will pass (SOA grade 6t), whil

Question

1000 candidates take SOA exam P. Countrywide, 35% will pass (SOA grade 6t), while 20% will just fail (SOA Grade 5). The others will get an SOA grade under 5, The country is made of three regions: East Coast, West Coast, and Central. The coasts each have 300 candidates taking the exam. 1, a. If the mark was independent of region, how many Central candidates would get a Actually, the marks may be dependent. The East Coast has a 25% pass rate, while two- b. Given a student passes, what's the probability the candidate was in the Central score under 5? thirds just fail (SOA grade 5). 50 candidates pass on the West Coast Region? Given a student does not pass, what's the probability the candidate was on the West Coast? c.

Explanation / Answer

P( grade 6+ ) = 0.35 and P( grade 5 ) = 0.2, therefore P( grade < 5) = 1 - P( grade 6+) - P( grade 5 ) that is:

P( grade < 5) = 1 - 0.35 - 0.2 = 0.45

P( east coast ) = 300/ 1000 because there are 300 candidates from east coast in total 1000 candidates.

P( east coast ) = 0.3 . Similarly P( west coast ) = 0.3 and therefore P( central ) = 1 - P( east coast ) - P( west coast )

that is P( central ) = 1 - 0.3 - 0.3 = 0.4

a) Here we are given that the marks were independent of the region, therefore

P(Central and grade < 5) = P( grade < 5) P(Central ) = 0.45*0.4 = 0.18

Therefore 0.18*1000 = 180 candidates in Central region would get a grade under 5

Next we are given that east coast pass rate is 25% which means that,

P( east coast and grade 6+) = 0.25*P(east coast) = 0.25*0.3 = 0.075

This means 0.075*1000 = 75 candidates pass from east coast.

Also we are given that 2/3 have a grade 5 in east coast and therefore

P( east coast and grade 5) = (2/3)*P(east coast) = (2/3)*0.3 = 0.2

Also we are given that P(west coast and 6+) = 50/1000 = 0.05 because 50 students pass in west coast.

Now as 75 students pass in east coast and 50 students pass on the west coast, also we know that the total number of students who pass is 0.35*1000 = 350, therefore the number of students who pass from central would be (350 - 50 - 75) = 225

b) Given a student pass, probability that the candidate is from central is computed as:

= Number of students from central who pass / Total number of passed students

= 225 / 350

= 0.6429

Therefore 0.6429 is the required probability here.

c) Given a student who does not pass, probability that the candidate was on the west coast is computed as:

= Total number of west coast candidates who do not pass / Total number of candidates who dont pass

= ( 300 - 50) / (1000 - 350)

= 250 / 650

= 0.3846

Therefore 0.3846 is the required probability here.

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