- SUPPOSE A JAR CONTAINS 15 RED MARBLES
AND 18 BLUE MARBLES.
IF YOU REACH IN THE JAR
AND PULL OUT TWO MARBLES AT RANDOM,
FIND THE PROBABILITY THAT BOTH ARE RED.
WE CAN FIND THIS PROBABILITY TWO WAYS.
ONE WAY WOULD BE TO THINK OF THIS AS SELECTING ONE MARBLE
AND THEN WITHOUT REPLACEMENT, SELECTING A SECOND MARBLE
AND FIND THE PROBABILITY THAT BOTH MARBLES ARE RED.
IF WE APPROACH IT THIS WAY,
WE WOULD USE THE PROBABILITY FORMULA GIVEN BELOW
WHEN EVENTS "A" AND B ARE NOT INDEPENDENT.
SO WE'LL FIRST FIND THE PROBABILITY THIS WAY
AND THEN FOR OUR SECOND APPROACH,
WE'LL THINK OF THIS AS PULLING OUT TWO MARBLES AT ONE TIME
AND USE COMBINATIONS TO FIND THE SAME PROBABILITY.
BUT TO BEGIN, WE'LL FIND THE PROBABILITY
OF SELECTING ONE MARBLE THAT'S RED
AND THEN ANOTHER MARBLE WITHOUT REPLACEMENT
THAT'S ALSO RED.
SO FROM OUR CONDITIONAL PROBABILITY FORMULA,
THIS IS EQUAL TO THE PROBABILITY
OF SELECTING ONE MARBLE THAT'S RED TIMES THE PROBABILITY
OF SELECTING A RED MARBLE
GIVEN A RED MARBLE HAS ALREADY BEEN SELECTED.
NOTICE HOW WE HAVE 15 + 18 OR 33 TOTAL MARBLES,
AND ON THE FIRST DRAW, 15 OF THEM WOULD BE RED.
SO THE PROBABILITY THAT THE FIRST MARBLE IS RED
WOULD BE 15 DIVIDED BY 33,
AND NOW WE HAVE TIMES THE PROBABILITY
THAT THE SECOND MARBLE IS RED
GIVEN WE KNOW THE FIRST MARBLE IS ALREADY RED.
WELL, THE FIRST THING TO RECOGNIZE
IS NOW THERE'S ONLY GOING TO BE 32 MARBLES
BECAUSE WE ALREADY SELECTED ONE RED MARBLE OUT
AND THERE'S ALSO ONLY GOING TO BE 14 RED MARBLES
SINCE WE ASSUME THE FIRST MARBLE IS ALREADY RED.
SO THE PROBABILITY OF RED GIVEN RED
WOULD BE 14 DIVIDED BY 32.
THIS PRODUCT WOULD BE OUR DESIRED PROBABILITY.
LET'S SIMPLIFY THESE FRACTIONS BEFORE MULTIPLYING.
NOTICE HOW 15 AND 33 SHARE A COMMON FACTOR OF THREE.
THERE IS FIVE THREE'S IN 15 AND 11 THREE'S IN 33.
14 AND 32 SHARE A COMMON FACTOR OF TWO.
THERE ARE SEVEN TWO'S IN 14 AND 16 TWO'S IN 32.
SO OUR PRODUCT IS 5/11 x 7/16,
WHICH WOULD BE 35 DIVIDED BY 176.
LET'S ALSO WRITE THE PROBABILITY
AS A DECIMAL AND PERCENTAGE.
TO CONVERT TO A DECIMAL, WE DIVIDE.
SO WE WOULD HAVE 35 DIVIDED BY 176.
LET'S ROUND TO FOUR DECIMAL PLACES.
NOTICE HOW THIS WOULD BE APPROXIMATELY 0.1989.
THE SIX INDICATES TO ROUND UP.
SO THIS IS AN APPROXIMATION, 0.1989.
AS A PERCENTAGE THIS WOULD BE 19.89%.
NOW, LET'S FIND THIS PROBABILITY
A SECOND WAY USING COMBINATIONS.
LET'S FIND THE PROBABILITY OF SELECTING TWO MARBLES
AT ONE TIME AND BOTH ARE RED.
SO WE WANT TO FIND THE PROBABILITY
OF SELECTING TWO REDS.
THE TOTAL NUMBER OF WAYS OF SELECTING TWO MARBLES,
BECAUSE THERE ARE 33 MARBLES, WOULD BE 33 CHOOSE 2
AND NOTICE HOW THERE ARE 15 RED MARBLES.
SO THE FAVORABLE NUMBER OF WAYS
OF SELECTING TWO RED MARBLES FROM 15 RED MARBLES
WOULD BE 15 CHOOSE 2.
WE CAN ALSO WRITE THESE COMBINATIONS
AS 15 CHOOSE TWO DIVIDED BY 33 CHOOSE 2.
THIS IS THE NOTATION OUR CALCULATOR USES.
AND WE ARE GOING TO EVALUATE THIS USING THE CALCULATOR
AND MAKE SURE WE DO GET THE SAME RESULT.
SO OUR NUMERATOR, IN PARENTHESIS,
WOULD BE 15 CHOOSE 2.
SO WE ENTER 15 MATH, ARROW OVER TO PROBABILITY
AND SELECT OPTION THREE FOR A COMBINATION
AND THEN WE ENTER 2.
SO THERE IS OUR NUMERATOR OF 15 CHOOSE 2
DIVIDED BY OUR DENOMINATOR OF 33 CHOOSE 2.
SO IN PARENTHESIS WE'LL HAVE 33 MATH PROBABILITY,
OPTION THREE AND THEN 2, CLOSED PARENTHESIS AND ENTER.
AND NOTICE HOW THIS DOES GIVE US THE SAME PROBABILITY
AS WE HAVE HERE.
IF WE PRESS MATH ENTER, ENTER,
IT WILL CONVERT TO A FACTION, 35/176.
SO IF YOU HAVEN'T LEARNED ABOUT COMBINATIONS,
WE SHOULD STILL BE ABLE TO DETERMINE THE PROBABILITY
USING THIS APPROACH HERE.
WE HAVE OUR EXACT PROBABILITIES HERE
AND WE HAVE THE APPROXIMATE PROBABILITIES GIVEN HERE,
AS WELL AS HERE.
I HOPE YOU FOUND THIS HELPFUL.