Freshmen at a college were classified according to whether their parents were bo

Question

Freshmen at a college were classified according to whether their parents were born in the United States and whether or not they were the first generation to attend college. This resulted in the accompanying table: Suppose that one of these students is selected at random. Define events A. B, and C as follows: A = the event that the selected student's parents were born in the United States B = the event that the selected student has had prior generations attend college C = the event that the selected student's parents were born somewhere else In the following table, indicate by putting an X in the appropriate cells which is true for each pair of events. It is not necessarily true that an X must go in a box for each row! Remember what the word disjoint means and how to prove independence!

Explanation / Answer

Let event A,B and C are,

A = the event that the selected student's parents were born in the United States.

B = the event that the selected student's has had prior generations attend college.

C = the event that the selected student's parents were born somewhere else.

We have given two by two contingency table.

First complete this table by summing all the rows and columns.

Now we have to define disjoint events and independent events.

Disjoint events: Two events that do not occur at the same time, also known as mutually exclusive events

Independent events: Unrelated events; the outcome of one event does not impact the outcome of the other event

Symbolically if A and B are two events then they are said to be disjoint

if P(A and B) = P(Phi) = 0

where Phi is the empty set.

And two events are said to be independent

if P(A and B) = P(A)*P(B)

Now we have given three pairs and we have to check if it is disjoint or independent.

First we check which are disjoint events.

For pair A, B :

P(A and B) = 24/100 = 0.24

A and B are not disjoint events.

For pair B, C :

P(B and C) = 6/100 = 0.06

B anc C are not disjoint events.

For pair A, C :

P(A and C) = 0

A and C are disjoint events.

P(A) = 0.8

P(B) = 0.3

P(C) = 0

Now we check independency for the events :

For pair A and B :

P(A and B) = P(A)*P(B)

0.24 = 0.8*0.3

0.24 = 0.24

LHS = RHS

A and B are independent events.

For pair B and C :

P(B and C) = P(B)*P(C)

0.06 = 0.3*0.2

0.06 = 0.06

LHS = RHS

B and C are independent events.

For pair A and C :

P(A and C) = P(A)*P(C)

0 = 0.8*0.2

0 = 0.16

LHS not= RHS

A and C are not independent events.

parents born here parents born else where total first generation to attend college 56 14 70 prior generation have attended college 24 6 30 total 80 20 100

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