Assume That you are taking two courses this semester (Math & Biology).
 The probability that you will pass Math is 0.835, the probability
that you will pass both courses is 0.276.  The probability that you
will pass at least one of the courses is 0.981.  (1) What is the
probability that you will pass the Bio course? (I think 0.146, or
0.981 - 0.835, is that logical or remotely correct) (2) Are the
passing of the two courses independent events? (justify using
probability) (3) are the events of passing the courses mutually
exclusive? why?

Hi engrintraining!!


1) What is the probability P(B) that you will pass the Bio course? 

If P(M) is the probability that you will pass Math we have that:

P(M or B) = P(B) + P(M) - P(M and B)

Then:

P(B) = P(M or B) - P(M) + P(M and B) =
     = 0.981 - 0.835 + 0.276 =
     = 0.422

The probability that you will pass the Bio course is 0.422


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(2) Are the passing of the two courses independent events? (justify using
probability)

Recall that when two events, M and B, are independent, the probability
of both occurring is:
  P(M and B) = P(M) * P(B)  

For this problem we know  that:
P(M) = 0.835
P(B) = 0.422
P(M and B) = 0.276

But P(M) * P(B) = 0.835 * 0.422 = 0.35237 that is different to P(M and
B) = 0.276 , then the passing of the two courses are not independent
events.


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(3) are the events of passing the courses mutually exclusive? why?

Two events are mutually exclusive if they cannot occur at the same
time. But the statement of this problem tell us that the probability
that you will pass both courses is not zero, then both events can
occur at the same time, so they are not mutually exclusive.
See this:
If M and B are mutually exclusive events, then: 

P(M or B) = P(M) + P(B), but:

P(M) + P(B) = 0.835 + 0.422 = 1.257 , then:

P(M or B) = P(M) + P(B) = 1.257 > 1  THIS IS AN ABSURD.

More yet:
We know that P(M or B) = 0.981 that is different to 1.257, this also
tell us that they are not mutually exclusive events.

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For aditional reference visit the following pages:
"Probability Definitions":
http://www.richland.cc.il.us/james/lecture/m116/sequences/probability.html

"Mutually Exclusive Events":
http://regentsprep.org/Regents/math/mutual/Lmutual.htm

"Independent Events":
http://regentsprep.org/Regents/math/mutual/Lindep.htm

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I hope that this helps you. 
Before rate this answer please feel free to request for a
clarification if you find something unclear, I will gladly respond
your requests for further assistance on this.


Best regards.
livioflores-ga

Thanks.  Very quick and detailed response.  Would describe this
researcher as excellent.