- WELCOME TO AN EXAMPLE OF THE COUNTING PRINCIPLE
AND PROBABILITY USING THE COUNTING PRINCIPLE.
WE'RE GIVEN THAT 3 BOYS AND 3 GIRLS STAND IN LINE.
THE FIRST QUESTION IS IN HOW MANY WAYS CAN THEY LINE UP?
ONE WAY TO ANSWER THIS QUESTION IS TO USE THE COUNTING PRINCIPLE
WHERE WE THINK OF THIS AS A SERIES OF EVENTS,
DETERMINE THE NUMBER OF WAYS EACH EVENT CAN OCCUR,
AND THEN FIND THE PRODUCT OF THESE NUMBERS OF WAYS.
AND SINCE THERE ARE 6 PEOPLE LINING UP,
WE'LL THINK OF THIS AS 6 DIFFERENT EVENTS,
AND WE'LL START BY DETERMINING THE NUMBER OF WAYS
WE CAN SELECT SOMEONE FOR THE FIRST POSITION.
WELL, SINCE THERE ARE 6 PEOPLE,
THERE ARE 6 WAYS TO SELECT THE PERSON FOR THE FIRST POSITION.
AND NOW FOR THE SECOND POSITION, SINCE THERE IS 5 PEOPLE LEFT,
THERE ARE 5 WAYS TO DETERMINE THE SECOND POSITION.
FOR THE THIRD POSITION THERE ARE 4 PEOPLE LEFT,
SO THERE ARE 4 WAYS TO SELECT SOMEONE FOR THIS POSITION
FOLLOWED BY 3 WAYS TO SELECT THE FOURTH POSITION.
THERE ARE 2 PEOPLE LEFT,
SO THERE ARE 2 WAYS TO SELECT SOMEONE FOR THE FIFTH POSITION
FOLLOWED BY 1 PERSON LEFT,
SO THERE'S 1 WAY TO SELECT SOMEONE FOR THE SIXTH POSITION.
SO THIS PRODUCT WOULD GIVE US A NUMBER OF WAYS
THAT 6 PEOPLE CAN LINE UP
WHICH WOULD BE 6 FACTORIAL WHICH = 720.
WE CAN ALSO THINK OF THIS AS A PERMUTATION
SINCE THE ORDER MATTERS 6 PERMUTES 6 WOULD ALSO = 720.
FOR PART B, WE'RE ASKED IN HOW MANY WAYS CAN THEY LINE UP
IF THE FIRST PERSON IN LINE IS A BOY,
AND THEN THE LINE ALTERNATES BY GENDER.
SO WE STILL HAVE 6 POSITIONS TO FILL IN LINE,
BUT NOW WE'RE TOLD THAT THE FIRST POSITION MUST BE A BOY.
AND THEN FROM HERE THE GENDER IS GOING TO ALTERNATE,
SO IT'LL GO BOY, GIRL, BOY, GIRL, BOY, GIRL.
AND SINCE THIS FIRST POSITION HAS TO BE A BOY
THERE'S ONLY 3 BOYS TO SELECT FROM,
SO THERE ARE 3 WAYS TO SELECT SOMEONE FOR THE FIRST POSITION.
THERE ARE ONLY 3 GIRLS,
SO THERE'S ONLY 3 WAYS TO SELECT THE SECOND POSITION.
THE THIRD POSITION MUST BE BOY.
THERE'S ONLY 2 BOYS REMAINING
BECAUSE ONE BOY IS IN THE FIRST POSITION.
SO THERE ARE 2 WAYS TO SELECT THE THIRD POSITION.
THE FOURTH POSITION MUST BE A GIRL.
THERE ARE 2 GIRLS REMAINING,
SO THERE ARE 2 WAYS TO SELECT A GIRL FOR THE FOURTH POSITION.
THERE'S ONLY 1 BOY REMAINING NOW
BECAUSE THERE'S A BOY IN POSITION 1 AND 3,
SO THERE'S ONLY 1 WAY TO SELECT A BOY FOR THE FIFTH POSITION
AND 1 WAY TO SELECT A GIRL FOR THE LAST POSITION.
SO THIS PRODUCT WILL BE THE NUMBER OF WAYS IN WHICH
A BOY IS FIRST AND THEN THE GENDER ALTERNATES.
WELL, 3 x 3 x 2 x 2 x 1 x 1 WOULD JUST BE 9 x 4 OR 36,
SO THERE ARE 36 WAYS THIS CAN OCCUR.
THE LAST 2 QUESTIONS DEAL WITH PROBABILITY.
PROBABILITY = THE NUMBER OF FAVORABLE WAYS
DIVIDED BY THE TOTAL NUMBER OF WAYS AN EVENT CAN OCCUR.
SO FOR PART C, WE WANT TO FIND THE PROBABILITY
THAT THE FIRST PERSON IN LINE IS A BOY,
AND THE LINE ALTERNATES BY GENDER.
WE CAN ACTUALLY FIND THIS PROBABILITY
USING THE ANSWERS FROM "A" AND B.
THE FAVORABLE NUMBER OF WAYS
THAT WE CAN HAVE BOY, GIRL, BOY, GIRL, BOY, GIRL
WE ALREADY FOUND FROM PART B THAT CAN OCCUR 36 WAYS,
THEN THE TOTAL NUMBER OF WAYS THAT THEY CAN LINE UP
IS FROM PART "A" WHICH IS 720 WAYS.
SO THIS IS THE PROBABILITY EXPRESSED AS A FRACTION.
LET'S GO AHEAD AND SIMPLIFY THIS.
THESE ARE BOTH DIVISIBLE BY 36.
AS A SIMPLIFIED FRACTION, THE PROBABILITY WOULD BE 1/20.
THE PROBABILITY CAN ALSO BE EXPRESSED AS A DECIMAL
OR A PERCENTAGE.
LET'S GO AHEAD AND CONVERT THIS TO A DECIMAL,
SO WE'D HAVE 1 DIVIDED BY 20
WHICH YOU PROBABLY RECOGNIZE AS 0.5.
SO WE CAN SAY THIS = 0.05
OR AS A PERCENTAGE, WE WOULD MULTIPLY BY 100
AND ADD A PERCENT SYMBOL, SO THIS WOULD BE 5%.
SO THESE 3 VALUES ARE EQUIVALENT TO EXPRESS
OUR PROBABILITY OF HAVING THE FIRST PERSON IN LINE
BE A BOY AND THEN ALTERNATING BY GENDER.
NOW FOR THE LAST QUESTION WE WANT TO FIND THE PROBABILITY
THAT THE GENDER ALTERNATES IN LINE,
MEANING WE CAN HAVE A BOY FIRST OR A GIRL FIRST,
BUT THEN THE GENDER MUST ALTERNATE.
SO WE CAN HAVE BOY, GIRL, BOY, GIRL, BOY, GIRL,
OR GIRL, BOY, GIRL, BOY, GIRL, BOY.
WELL, WE ALREADY KNOW THE FAVORABLE NUMBER OF WAYS
OF HAVING A BOY FIRST
AND THEN ALTERNATING BY GENDER IS 36 FROM PART C OR PART B,
BUT NOTICE HOW IF WE HAD GIRL, BOY, GIRL, BOY, GIRL, BOY,
THE NUMBER OF WAYS THIS CAN OCCUR WOULD ALSO BE 36.
SO WE'LL HAVE TO ADD ANOTHER 36 FAVORABLE WAYS
DIVIDED BY THE TOTAL NUMBER OF WAYS WHICH IS STILL 720.
SO NOW OUR PROBABILITY IS 72 DIVIDED BY 720
WHICH SIMPLIFIES NICELY TO 1/10.
NOTICE HOW THE PROBABILITY IS DOUBLE THE PROBABILITY
OF HAVING A BOY FIRST AND THEN ALTERNATING GENDER.
SO AS A DECIMAL THIS WOULD BE 0.1.
AS A PERCENT THIS WOULD BE 10%.
OKAY, THAT'S GOING TO DO IT FOR THIS EXAMPLE.
I HOPE YOU FOUND THIS HELPFUL.