- WELCOME TO AN INTRODUCTION TO PROBABILITY.
THE PROBABILITY OF AN EVENT
REPRESENTS THE LIKELIHOOD IT WILL OCCUR.
THE PROBABILITY COMPARES THE FAVORABLE NUMBER OF OUTCOMES
TO THE TOTAL NUMBER OF OUTCOMES AS WE SEE HERE.
AND THE PROBABILITY OF AN EVENT CAN BE EXPRESSED
AS A FRACTION, DECIMAL, OR PERCENTAGE.
FOR EXAMPLE, OFTEN WHEN WE LISTEN TO THE WEATHER FORECAST
THEY GIVE US THE PERCENT CHANCE THAT IT WILL RAIN,
WHICH REPRESENTS THE PROBABILITY THAT IT WILL RAIN.
FOR EXAMPLE, IF THERE IS A 20% CHANCE OF RAIN
THAT MEANS THERE ARE 20 CHANCES OUT OF 100
WRITTEN IN FRACTION FORM, WHICH SIMPLIFIES TO 1/5,
WHICH CAN ALSO BE WRITTEN AS 0.2 OR 2/10.
SO ALL OF THESE ARE EQUAL TO EACH OTHER
AND THEY ALL REPRESENT THE PROBABILITY,
IN THIS CASE, THAT IT WOULD RAIN.
NOW, THERE ARE TWO TYPES OF PROBABILITY.
THEORETICAL PROBABILITY IS FOUND BY USING MATHEMATICS
AND EXPERIMENTAL PROBABILITY IS FOUND
BY ACTUALLY CONDUCTING AN EXPERIMENT.
THE TOTAL NUMBER OF OUTCOMES IS OFTEN CALLED THE SAMPLE SPACE
AND THE FAVORABLE NUMBER OF OUTCOMES
IS OFTEN CALLED THE EVENT.
LET'S GO AHEAD AND TAKE A LOOK AT A PROBLEM.
LET'S SAY WE HAVE A BAG OF MARBLES
THAT CONTAINS THE NUMBERS ONE THROUGH EIGHT,
THEN THE SAMPLE SPACE WOULD BE EQUAL TO THE NUMBERS
ONE THROUGH EIGHT.
FOR NUMBER ONE, IF WE WANT TO KNOW WHAT THE PROBABILITY IS
OF SELECTING A TWO FROM THE BAG,
WELL THE ONLY SUCCESSFUL EVENT WOULD BE SELECTING A TWO
FROM THE EIGHT NUMBERS.
SO THE PROBABILITY OF SELECTING A 2 FROM THE BAG
WOULD BE 1 FAVORABLE OUTCOME
OVER A TOTAL OF 8 POSSIBLE OUTCOMES OR 1/8,
WHICH AGAIN COULD BE LEFT AS A FRACTION
OR IF WE CONVERT THIS TO A DECIMAL BY TAKING 1
AND DIVIDING BY 8 WE WOULD HAVE 0.125,
WHICH IS ALSO EQUAL TO 12.5%.
NUMBER TWO ASKS,
WHAT IS THE PROBABILITY OF SELECTING AN ODD NUMBER?
WELL, NOW THE FAVORABLE NUMBER OF OUTCOMES
WOULD BE ONE, THREE, FIVE, AND SEVEN.
WELL, THE SAMPLE SPACE HERE STAYS THE SAME.
SO THE PROBABILITY OF SELECTING AN ODD NUMBER,
THERE ARE 4 FAVORABLE OUTCOMES OUT OF A TOTAL OF 8 AGAIN,
WHICH IS EQUAL TO 1/2, .5 OR 50%.
AND THEN FOR NUMBER THREE,
WHAT IS THE PROBABILITY OF SELECTING A NUMBER >6?
WELL, THE SUCCESSFUL OUTCOMES WOULD BE >6 WOULD MEAN 7 OR 8
AS A SUCCESSFUL OUTCOME.
SO THE PROBABILITY OF SELECTING A NUMBER THAT'S >6
WOULD BE 2 FAVORABLE OUTCOMES OUT OF A TOTAL OF 8 AGAIN,
WHICH OF COURSE SIMPLIFIES TO 1/4.
AND IN DECIMAL FORM THIS WOULD BE 0.25, WHICH IS EQUAL TO 25%.
IF THE PROBABILITY OF AN EVENT IS P OF E,
THEN THE PROBABILITY OF THE COMPLIMENT
WOULD BE THE PROBABILITY THAT THE EVENT E WOULD NOT HAPPEN.
ONE THING WE SHOULD REALIZE IS IF WE CONSIDER THE PROBABILITY
THAT AN EVENT WILL OCCUR
AND THEN THE PROBABILITY THAT THE EVENT WILL NOT OCCUR,
THAT SUM SHOULD EQUAL 1 REPRESENTING 100%.
THERE'S A 100% CHANCE THAT AN EVENT WILL OCCUR OR NOT OCCUR.
SO IF WE WANT TO KNOW THE PROBABILITY
THAT AN EVENT WILL NOT OCCUR THEN WE CALL THAT E PRIME.
THE PROBABILITY OF E PRIME WOULD BE EQUAL TO ONE
- THE PROBABILITY THAT EVENT WILL OCCUR.
AND HERE'S AN EXAMPLE OF WHAT WE'RE TALKING ABOUT HERE.
WHAT IS THE PROBABILITY OF ROLLING A TWO
ON A STANDARD DIE WITH SIX SIDES?
SO REMEMBER THAT A DIE IS JUST A CUBE
THAT HAS THE NUMBERS ONE THROUGH SIX
WITH ONE NUMBER ON EACH SIDE.
SO THERE ARE SIX POSSIBLE OUTCOMES
WHEN WE ROLL A STANDARD DIE.
THE TOTAL NUMBER OF WAYS TO ROLL A 2 WOULD BE 1 WAY OUT OF 6
OR 1/6 WOULD BE THE PROBABILITY.
FOR THE PROBABILITY THAT WE DO NOT ROLE A TWO
COULD BE FOUND TWO WAYS.
THE NUMBERS ONE THROUGH SIX THAT ARE NOT A TWO
WOULD BE ONE, THREE, FOUR, FIVE, AND SIX.
SO THAT WOULD GIVE US FIVE FAVORABLE OUTCOMES
OUT OF A TOTAL OF SIX.
5/6, OR USING THIS IDEA HERE
SINCE WE ALREADY FOUND THE PROBABILITY OF ROLLING A 2,
WE COULD JUST SAY THE PROBABILITY OF NOT A 2
WOULD BE 1 - THE PROBABILITY OF A 2. 1 - 1/6 = 5/6 AS WELL.
SO THAT'S THE IDEA OF A COMPLEMENT
OF A GIVEN PROBABILITY.
LET'S GO AHEAD AND TAKE A LOOK AT ONE MORE PROBLEM.
HERE WE HAVE A SITUATION WITH A STANDARD DECK OF CARDS.
IF YOU'RE NOT FAMILIAR WITH A STANDARD DECK OF CARDS,
I'VE DESCRIBED HERE WHAT MAKES UP A DECK OF 52 CARDS.
SO THESE PROBLEMS WE'RE ASSUMING
WE'RE SELECTING ONE CARD FROM THE DECK.
AND SO FOR NUMBER ONE WE WANT TO KNOW
WHAT'S THE PROBABILITY OF A FACE CARD?
WELL, THE TOTAL NUMBER OF OUTCOMES WOULD BE 52
BECAUSE THERE ARE 52 CARDS IN THE DECK.
THE NEXT THING WE HAVE TO FIGURE OUT
IS HOW MANY FACE CARDS ARE THERE.
WELL, JACK, QUEEN, AND KINGS ARE THE FACE CARDS,
BUT THERE ARE FOUR JACKS, FOUR QUEENS, AND FOUR KINGS
BECAUSE THERE ARE FOUR SUITS.
SO THAT WOULD BE A TOTAL OF 12 FACE CARDS.
SO THE PROBABILITY OF SELECTING ONE CARD FROM A DECK
AND IT BEING A FACE CARD WOULD BE 12/52.
THESE DO HAVE A COMMON FACTOR OF 4 SO THIS WOULD BE 3/13.
AND AGAIN, WE COULD EXPRESS THIS AS A DECIMAL OR A PERCENTAGE,
BUT WE'LL GO AHEAD AND LEAVE IT AS A FRACTION.
SO FOR NUMBER TWO, IF WE SELECT ONE CARD FROM A DECK
WE WANT TO KNOW THE PROBABILITY OF IT BEING A FIVE.
WELL, AGAIN, THERE'S 52 CARDS.
THAT'S THE SAMPLE SPACE.
AND THERE ARE FOUR FIVES, THE FIVE OF HEARTS,
FIVE OF DIAMONDS, FIVE OF CLUBS, AND FIVE OF SPADES.
SO THAT WOULD BE 4/52.
AGAIN, WE HAVE A COMMON FACTOR OF 4 SO THIS GIVES US 1/13.
THE PROBABILITY OF SELECTING A FIVE FROM THE DECK
WOULD BE 1/13.
AND THEN FOR NUMBER THREE NOTICE
WE WANT THE PROBABILITY OF SELECTING A CARD
THAT WOULD BE A NON-FACE CARD.
AGAIN, THERE'S TWO WAYS TO DO THIS.
SINCE WE ALREADY FOUND THE PROBABILITY
OF SELECTING A FACE CARD FROM THE DECK,
THE PROBABILITY OF PICKING A NON-FACE CARD IS THE COMPLEMENT.
AND THEREFORE, IT WOULD BE EQUAL TO 1 - THE PROBABILITY
OF A FACE CARD, WHICH IS EQUAL TO 1 - 3/13.
WELL, 1 IS EQUAL TO 13/13 SO THIS WOULD BE 10/13.
OF COURSE, THE OTHER WAY TO DO IT
IS JUST CONSIDER HOW MANY NON-FACE CARDS ARE THERE.
WELL, THERE'S 52 CARDS AND F THERE'S 12 FACE CARDS,
52 - 12 WOULD LEAVE 40 NON FACE CARDS.
AND IF WE SIMPLIFY THIS FRACTION
WE ARE GOING TO GET 10/13 AS WELL.
THAT'LL DO IT FOR THIS INTRODUCTION VIDEO.
IN THE NEXT VIDEO WE'LL TAKE A LOOK
AT SOME MORE INVOLVED PROBABILITY QUESTIONS
THAT ALSO INVOLVE COMBINATIONS.
SO I WONDER WHAT THE PROBABILITY IS
THAT YOU'LL WATCH THE NEXT VIDEO.