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- A GROUP OF STUDENTS TOOK A TEST.
THE GRADES AND GENDER ARE SUMMARIZED BELOW.
IF ONE STUDENT IS CHOSEN AT RANDOM
FIND THE PROBABILITY THAT THE STUDENT WAS FEMALE
OR GOT A C ON THE TEST.
SO WE WANT TO FIND THE PROBABILITY
THAT IF WE SELECT ONE STUDENT IT WOULD BE A FEMALE
OR THE STUDENT WOULD HAVE A C ON THE TEST.
THESE TWO EVENTS CAN OCCUR AT THE SAME TIME
SINCE THERE ARE SOME FEMALES
THAT ALSO RECEIVED A C ON THE TEST,
AND THEREFORE THESE EVENTS ARE NOT MUTUALLY EXCLUSIVE,
AND THEREFORE,
WE'LL HAVE TO APPLY THIS PROBABILITY FORMULA HERE
FOR THE PROBABILITY OF THE UNION OF TWO EVENTS
WHEN THEY ARE NOT MUTUALLY EXCLUSIVE.
THIS PROBABILITY IS EQUAL
TO THE PROBABILITY OF SELECTING A FEMALE
+ THE PROBABILITY OF A STUDENT RECEIVING A C
- THE PROBABILITY OF A STUDENT BEING A FEMALE
AND RECEIVING A C.
WE'LL BE ABLE TO TELL FROM THE TABLE
WHY YOU HAVE TO SUBTRACT THIS PROBABILITY HERE.
NOTICE FROM THE TABLE, LOOKING AT THE TOTAL,
65 STUDENTS TOOK THE TEST AND NOTICE THIS ROW HERE
GIVES US INFORMATION ABOUT THE FEMALES.
NOTICE HOW THERE WERE A TOTAL OF 25 FEMALES THAT TOOK THE TEST
AND THEREFORE
THE PROBABILITY OF SELECTING A FEMALE FROM THIS GROUP
WOULD BE EQUAL TO 25 DIVIDED BY 65 +--
NOW WE WANT THE PROBABILITY THAT A STUDENT SELECTED AT RANDOM
WOULD HAVE A C ON THE TEST.
THAT INFORMATION IS GIVEN IN THIS COLUMN HERE.
NOTICE HOW 12 OF THE STUDENTS IN THE CLASS RECEIVED A C.
AND THEREFORE,
THE PROBABILITY OF SELECTING A STUDENT THAT RECEIVED A C
WOULD BE 12/65 OR 12 DIVIDED BY 65.
BUT NOW, NOTICE HOW WE COUNTED THESE 9 FEMALE STUDENTS
THAT ALSO RECEIVED A C,
THESE STUDENTS HERE IN BOTH PROBABILITIES
AND THAT'S THE REASON WHY WE HAVE TO SUBTRACT
THE PROBABILITY OF A STUDENT BEING A FEMALE AND A C.
WE CAN'T COUNT THEM TWICE TO GET THE CORRECT PROBABILITY.
SO THE PROBABILITY OF A STUDENT BEING A FEMALE AND RECEIVING A C
WOULD BE EQUAL TO 9 DIVIDED BY 65.
SO THE PROBABILITY OF ONE STUDENT BEING SELECTED
AND THE STUDENT BEING A FEMALE
OR RECEIVING A C WOULD BE EQUAL TO 25/65 + 12/65 - 9/65.
NOTICE HOW WE HAVE A COMMON DENOMINATOR OF 65
AND THE NUMERATOR IS 25 + 12 WHICH IS 37 - 9 WHICH IS 28.
WHICH DOES NOT SIMPLIFY SO THIS WOULD BE OUR EXACT PROBABILITY.
LET'S ALSO EXPRESS THIS AS A DECIMAL AND PERCENTAGE.
TO CONVERT TO A DECIMAL WE'LL DIVIDE.
WE HAVE 28 DIVIDED BY 65. LET'S ROUND TO 4 DECIMAL PLACES.
THIS WILL BE APPROXIMATELY 0.4308,
THE 6 INDICATES WE ROUND UP.
AS A PERCENTAGE, THIS WOULD BE 43.08%.
SO JUST REMEMBER THAT THIS IS THE EXACT PROBABILITY
AND THESE TWO HAVE BEEN ROUNDED SO THEY'RE APPROXIMATIONS.
I HOPE YOU FOUND THIS HELPFUL.