- WELCOME TO TWO EXAMPLES OF DETERMINING PROBABILITY
OF EVENTS THAT ARE NOT MUTUALLY EXCLUSIVE.
IF IT IS POSSIBLE FOR TWO EVENTS TO OCCUR AT THE SAME TIME,
THE EVENTS ARE NOT MUTUALLY EXCLUSIVE,
SO IF EVENTS "A" AND B ARE NOT MUTUALLY EXCLUSIVE
THEN THE PROBABILITY OF "A" OR B = THE PROBABILITY OF "A"
+ THE PROBABILITY OF B - THE PROBABILITY OF "A" AND B.
FROM OUR TWO EXAMPLES WE'LL SEE
WHY WE HAVE TO SUBTRACT THIS PROBABILITY HERE.
SO FIRST, ONE CARD IS SELECTED FROM A DECK OF PLAYING CARDS.
WHAT IS THE PROBABILITY IT IS A CLUB OR A FACE CARD?
SO BECAUSE A CARD CAN BE BOTH A CLUB AND A FACE CARD
THESE TWO EVENTS ARE NOT MUTUALLY EXCLUSIVE
AND THEREFORE TO FIND THIS PROBABILITY
WE'LL FIND THE PROBABILITY THAT THE CARD IS A CLUB,
AND THEN WE'LL ADD THE PROBABILITY OF A FACE CARD,
BUT THEN WE'LL SUBTRACT THE PROBABILITY
THE CARD IS A CLUB AND A FACE CARD.
SO REMEMBER, OUR DECK OF CARDS HAS 52 CARDS.
THERE WOULD BE 13 CLUBS. THAT WOULD BE THIS SUIT HERE.
AND THERE ARE 12 FACE CARDS.
WE HAVE THE JACK, QUEEN, AND KING OF ALL THREE SUITS,
AND THEREFORE THE PROBABILITY OF A CLUB WOULD BE 13 OUT OF 52
+ THE PROBABILITY OF A FACE CARD,
THAT WOULD BE 12 OUT OF 52.
BUT NOW WE HAVE TO SUBTRACT THE PROBABILITY THAT THE CARD
WOULD BE A CLUB AND A FACE CARD.
THE REASON WE HAVE TO DO THIS IS
THAT NOTICE THAT THE CARDS THAT ARE BOTH CLUBS AND FACE CARDS,
OR THESE THREE CARDS HERE
WERE INCLUDED IN THE FAVORABLE NUMBER OF WAYS
FOR BOTH THE CLUBS AND THE FACE CARDS,
SO THESE THREE CARDS WERE COUNTED TWICE,
AND THEREFORE WE HAVE TO SUBTRACT THIS PROBABILITY HERE
WHICH WOULD BE 3 OUT OF 52.
SO WE HAVE A COMMON DENOMINATOR OF 52.
THE NUMERATOR IS GOING TO BE 13 + 12, THAT'S 25 - 3. THAT'S 22.
NOW, IF WE STOP HERE FOR A MOMENT,
NOTICE HOW THIS IS TELLING US THERE ARE 22 FAVORABLE WAYS
TO PICK A CARD THAT IS A CLUB OR A FACE CARD OUT OF 52.
SO IF WE WERE TO CLEAR THIS FOR A MOMENT,
IF WE IDENTIFIED THE CARDS THAT WERE CLUBS OR FACE CARDS
AND ONLY COUNTED THEM ONCE THAT WOULD BE THESE CARDS HERE,
AND THAT WOULD BE 22 CARDS OUT OF 52,
BUT THIS FRACTION DOES SIMPLIFY WITH A COMMON FACTOR OF 2.
THIS SIMPLIFIES TO 11/26.
LET'S GO AHEAD AND GIVE THIS AS A DECIMAL AS WELL.
11 DIVIDED BY 26 WOULD GIVE US THE DECIMAL VALUE,
SO IT'S APPROXIMATELY 0.4231.
SO AS A PERCENTAGE WE WOULD MULTIPLY BY 100,
ADD A PERCENT SYMBOL, THAT'S APPROXIMATELY 42.31%.
TO CONVERT THIS TO A PERCENTAGE WE WOULD MULTIPLY BY 100,
ADD A PERCENT SYMBOL
OR MOVE THE DECIMAL POINT TO THE RIGHT TWO PLACES.
THAT WOULD BE 42.31%. LET'S TAKE A LOOK AT OUR SECOND EXAMPLE.
20 PING-PONG BALLS ARE NUMBERED FROM 1 TO 20.
WHAT IS THE PROBABILITY OF SELECTING 1 BALL
AND IT IS ODD OR LESS THAN 5?
AGAIN, THESE 2 EVENTS ARE NOT MUTUALLY EXCLUSIVE
BECAUSE THE BALL CAN BE ODD AND LESS THAN 5 AT THE SAME TIME,
AND THEREFORE OUR PROBABILITY WILL BE EQUAL TO THE PROBABILITY
OF THE BALL BEING ODD
+ THE PROBABILITY THE BALL IS LESS THAN 5
- THE PROBABILITY THAT THE BALL IS ODD AND LESS THAN 5.
WELL, IF WE CONSIDER THE NUMBERS FROM 1 TO 20,
10 OF THESE NUMBERS WOULD BE ODD.
1, 3, 5, 7, 9, 11, 13, AND SO ON.
SO THE PROBABILITY THE BALL WOULD BE ODD WOULD BE 10/20
+ THE PROBABILITY THAT THE BALL WOULD BE LESS THAN 5.
WELL, THE BALLS THAT ARE LESS THAN 5 WOULD BE 1 THROUGH 4,
SO THE PROBABILITY OF BEING LESS THAN 5 WOULD BE 4/20,
BUT NOTICE HOW WE COUNTED THE NUMBER 3 AND THE NUMBER 1
IN BOTH PROBABILITIES
AND THEREFORE NOW WE HAVE TO SUBTRACT THE PROBABILITY
THAT THE BALL WOULD BE BOTH ODD AND LESS THAN 5
WHICH WOULD BE WHEN THE BALL IS 1 AND 3,
THEREFORE WE HAVE TO SUBTRACT 2 OUT OF 20 OR 2/20.
SO WE HAVE 20 AS A DENOMINATOR,
AND THEN WE HAVE 10 + 4, THAT'S 14 - 2. THAT'S 12.
COMMON FACTOR OF 4.
SO WE HAVE 3/5 WHICH = 3 DIVIDED BY 5 OR 0.6 OR 60%.
BUT NOTICE HOW IF WE JUST STOP AND FOCUS ON 12/20 FOR A MOMENT,
IF WE WERE TO CLEAR OUR WORK BELOW
AND IDENTIFY THE NUMBERS THAT ARE ODD OR LESS THAN 5,
IF WE'RE STARTING AT 1, 1 IS ODD, 2 IS LESS THAN 5,
3 IS ODD, 4 IS LESS THAN 5, 5 IS ODD, 6 IS NOT ODD,
AND IT'S ALSO NOT LESS THAN 5,
SO FROM HERE WE JUST SELECT THE ODDS
SO 7, 9, 11, 13, 15, 17, 19.
NOTICE HOW THERE ARE 5, 6, 7, 8,
9, 10, 11, 12 FAVORABLE WAYS OUT OF 20
WHICH WE HAVE HERE USING THE FORMULA TO SELECT A BALL
THAT IS ODD OR LESS THAN 5.
BECAUSE THESE TWO EXAMPLES ARE PRETTY STRAIGHTFORWARD,
WE COULD PROBABLY JUST COUNT THE FAVORABLE NUMBER OF WAYS
THAT THESE EVENTS COULD OCCUR.
IT'S ALSO A NICE WAY TO UNDERSTAND
WHY WE HAVE THIS SPECIAL FORMULA
WHEN WE HAVE EVENTS THAT ARE NOT MUTUALLY EXCLUSIVE.
I HOPE YOU FOUND THESE TWO EXAMPLES HELPFUL.