- GIVING A TEST TO A GROUP OF STUDENTS,
THE GRADES AND GENDER ARE SUMMARIZED BELOW.
IF ONE STUDENT IS CHOSEN AT RANDOM,
FIND THE PROBABILITY THAT THE STUDENT DID NOT GET A B.
LET'S FIRST FIND THIS PROBABILITY
USING THE BASIC PROBABILITY FORMULA,
AND THEN WE'LL USE THE FORMULA FOR THE COMPLEMENT OF AN EVENT.
SO USING THE BASIC PROBABILITY FORMULA
THE PROBABILITY OF A STUDENT NOT HAVING A GRADE OF B
WOULD BE EQUAL TO THE FAVORABLE NUMBER OF OUTCOMES
DIVIDED BY THE TOTAL NUMBER OF OUTCOMES,
OR IN THIS CASE IT WOULD BE
THE TOTAL NUMBER OF STUDENTS NOT GETTING A B
DIVIDED BY THE TOTAL NUMBER OF STUDENTS TAKING THE TEST.
WELL, LOOKING AT THE TABLE, IF A STUDENT DID NOT GET A B,
THEY EITHER RECEIVED AN "A" OR A C.
NOTICE 28 STUDENTS RECEIVED AN "A," WHICH IS NOT A B,
AND 23 STUDENTS RECEIVED A C WHICH IS NOT A "B."
AND THEN FINALLY A TOTAL OF 89 STUDENTS TOOK THE TEST.
SO THE PROBABILITY THAT A STUDENT DID NOT GET A B
WOULD BE EQUAL TO 28 + 23,
THE FAVORABLE NUMBER OF OUTCOMES OR THE NUMBER OF GRADES
THAT WERE NOT A B DIVIDED BY THE TOTAL NUMBER OF STUDENTS
TAKING THE TEST WHICH IS 89.
WHICH GIVES US 51 DIVIDED BY 89 FOR THE PROBABILITY
A RANDOM STUDENT DID NOT GET A B.
PROBABILITY CAN BE EXPRESSED
AS A FRACTION, DECIMAL, OR PERCENTAGE.
LET'S LEAVE THIS AS A FRACTION FOR RIGHT NOW,
BUT NOTICE HOW IF THIS DID SIMPLIFY WE SHOULD,
BUT THIS DOES NOT SIMPLIFY.
NOW LET'S USE THE COMPLEMENT FORMULA GIVEN HERE.
THE PROBABILITY OF NOT B
WHICH WE COULD ALSO WRITE AS B WITH A BAR OVER IT
IS ALSO EQUAL TO 1 - THE PROBABILITY OF A B
WHICH WOULD BE
1 - THE PROBABILITY OF A STUDENT RECEIVING A B
WOULD BE EQUAL TO THE NUMBER OF STUDENTS THAT RECEIVED A B
DIVIDED BY THE TOTAL NUMBER OF STUDENTS THAT TOOK THE TEST.
WELL, NOTICE THAT 38 STUDENTS RECEIVED A B,
THEREFORE THE PROBABILITY THAT A STUDENT RECEIVED A B
WOULD BE 38 DIVIDED BY 89. AND NOW LET'S FIND THIS DIFFERENCE.
1 = 89/89. AGAIN, NOTICE HOW THIS = 1,
BUT NOW WE HAVE A COMMON DENOMINATOR,
SO WE CAN GO AHEAD AND SUBTRACT.
NOTICE HOW THIS DOES GIVE US 51/89.
SO OF COURSE THE PROBABILITY IS THE SAME
WHETHER I USE THE BASIC PROBABILITY FORMULA
OR THE PROBABILITY FOR THE COMPLEMENT OF AN EVENT.
LET'S ALSO TAKE THE TIME TO WRITE THIS PROBABILITY
AS A DECIMAL AND A PERCENTAGE.
TO CONVERT A FRACTION TO A DECIMAL, WE WOULD DIVIDE,
SO WE'D HAVE 51 DIVIDED BY 89.
LET'S ROUND TO 4 DECIMAL PLACES,
SO THIS WOULD BE APPROXIMATELY 0.5730.
NOTICE HOW THE 3 HERE INDICATES TO ROUND DOWN.
AS A PERCENTAGE WE WOULD MULTIPLY BY 100
AND ADD A PERCENT SIGN
OR MOVE THE DECIMAL POINT TO THE RIGHT TWO TIMES
GIVING US 57.30%.
JUST KEEP IN MIND THE FRACTIONAL VALUE IS EXACT,
AND THE DECIMAL AND PERCENTAGE ARE APPROXIMATIONS.
I HOPE YOU FOUND THIS HELPFUL.