Question

Using conditional probability, can someone please help me answer this in laymans terms to understand this.

Question 2 Public vs Private School The Private School Association conducts a survey of the outcome of private and public school students who pursue university education. The outcome is either graduate eventually or not. It is found that: • 60% of commencing freshmen are from public schools. • 75% of public student students who commence their university education eventually graduate. . 90% of private school freshmen eventually graduate. Is there any evidence that a freshman's chance to graduate is dependent on the type of high school they attended?

Answer 1

Let us denote the events

A : a randomly selected student is from public school

B :  a randomly selected student commence his university education

Given

$P(A) = 0.6, P(B|A) = 0.75, P(B|A^c) = 0.9$

That implies $P(A^c) = 1 - P(A)= 1 - 0.6 = 0.4$

Now,

$P(B) = P(B|A)*P(A) + P(B|A^c)*P(A^c)$

$\Rightarrow P(B) = (0.75*0.6)+(0.9*0.4)$

$\Rightarrow P(B) = 0.81$

Since $P(B|A) \neq P(B)$ , so we can say that freshman's chance to graduate is not independent of the type of high school they attended.