A college student is taking two courses The probability she

A college student is taking two courses.
The probability she passes the first course is 0.76.
The probability she passes the second course is 0.67.
The probability she passes at least one of the courses is 0.91.

Give answers to four decimal places

I tried the question and got everything wrong except the true or false question. My professor said I need to use conditional probabilities or complements to answer the question since the second question is false (not independent)

PARTS E, F, G are the only ones I need answered, the rest were answered in another post. Thanks!

a) What is the probability she passes both courses?

b) Is the event she passes one course independent of the event that she passes the other course?

TRUE OR FALSE ---> I know it\'s false.

c) What is the probability she does not pass either course (has two failing grades)?

d) What is the probability she does not pass both courses (does not have two passing grades)?

e) What is the probability she passes exactly one course?

f) Given she passes the first course, what is the probability she passes the second?

g) Given she passes the first course, what is the probability she does not pass the second?

Solution

Let

A = pass first course
B = pass second course

Hence,

a)

P(A and B) = P(A) + P(B) - P( A U B) = 0.76 + 0.67 - 0.91 = 0.52 [answer]

b)

If they are independent,

P(A|B) = P(A)

As

P(A|B) = P(A and B) / P(B) = 0.52/0.67 = 0.776119403
P(A) = 0.76

Then no, they are NOT INDEPENDENT.

*****************

c)

P(A U B)\' = 1 - P(A U B) = 1 - 0.91 = 0.09 [answer]

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d)

P(A and B)\' = 1 - P(A and B) = 1 - 0.52 = 0.48 [answer]

e)

P(A only or B only) = P(A) + P(B) - P(A and B)

= 0.76+0.67-2*0.52 = 0.39 [answer]

********************

f)

P(B|A) = P(A and B) / P(A)

=0.52/0.76 = 0.684210526 [answer]

**************

g)

P(B\'|A) = 1 - P(B|A)

=1-0.684210526

=0.315789474 [answer]