Ex 1:  Probability Using Combinations (Lottery)

Ex 1: Probability Using Combinations (Lottery)


– WELCOME TO AN EXAMPLE
ON PROBABILITY
THAT INVOLVES THE USE
OF COMBINATIONS.
A STATE LOTTERY IS SET UP
SO EACH PLAYER
CHOOSES 6 DIFFERENT NUMBERS
FROM 1 THROUGH 44.
WHAT IS THE PROBABILITY
OF WINNING WITH ONE TICKET?
WHAT IS THE PROBABILITY
OF WINNING WITH 2,000 TICKETS?
TO WIN, YOU HAVE TO SELECT
ALL SIX NUMBERS CORRECTLY.
FOR REFERENCE, BELOW WE HAVE
THE DEFINITION OF PROBABILITY
AND THE DEFINITION
OF A COMBINATION.
TO FIND THE PROBABILITY,
WE WANT TO FIND THE NUMBER
OF FAVORABLE WAYS
AND DIVIDE BY THE TOTAL NUMBER
OF WAYS THE EVENT CAN OCCUR.
SO TO FIND THE PROBABILITY
OF WINNING WITH ONE TICKET–
WELL, IF WE ONLY HAVE 1 TICKET
THERE’S ONLY 1 CHANCE
OR 1 FAVORABLE WAY OF WINNING
THE LOTTERY,
SO WE’LL HAVE 1 DIVIDED BY
THE TOTAL NUMBER OF WAYS
OF SELECTING 6 NUMBERS
FROM 44 NUMBERS
WHICH WOULD BE A COMBINATION,
WHICH WOULD BE 44 CHOOSE 6.
 
NOTICE HOW THERE ARE SEVERAL
WAYS TO WRITE A COMBINATION.
NOW WE’LL EVALUATE
THIS COMBINATION BY HAND
AS WELL AS ON THE CALCULATOR.
LET’S START BY DOING IT BY HAND.
IF WE HAVE N CHOOSE R
THIS=N FACTORIAL DIVIDED BY
N – R FACTORIAL, R FACTORIAL
SO LET’S GO AHEAD AND EVALUATE
THIS BY HAND FIRST.
WE WOULD HAVE 44 FACTORIAL
DIVIDED BY 44 – 6 FACTORIAL.
THAT WOULD BE 38 FACTORIAL
AND THEN 6 FACTORIAL.
SO NOW WE’RE GOING TO GO AHEAD
AND EXPAND 44 FACTORIAL,
BUT THEN WE’LL STOP ONCE
WE REACH 38
AND WRITE IT AS 38 FACTORIAL.
SO WE’LL HAVE 44 x 43 x 42 x 41
x 40 x 39 x 38 AND SO ON
WHICH AGAIN WOULD BE
38 FACTORIAL.
THE DENOMINATOR IS 38 FACTORIAL.
LET’S GO AHEAD
AND EXPAND 6 FACTORIAL.
THAT WOULD BE 6 x 5 x 4
x 3 x 2 x 1.
SO NOW 38 FACTORIAL/38 FACTORIAL
SIMPLIFIES TO 1,
AND THEN NOTICE THAT 5 x 4
WOULD BE 20 x 2 WOULD BE 40.
40/40 SIMPLIFIES TO 1, AND THEN
3 DIVIDES EVENLY INTO 39.
THERE’S ONE 3 IN 3
AND 13 3’s IN 39.
6 DIVIDES EVENLY INTO 42.
SINCE 6 x 7 IS 42 THERE IS ONE 6
IN 6, AND SEVEN 6’s IN 42.
NOW THE DENOMINATOR IS 1,
WE CAN FIND THE PRODUCT
IN THE NUMERATOR
WHICH WILL GIVE US THE VALUE
OF 44 CHOOSE 6.
44 x 43 x 7 x 41 x 13.
WE HAVE 7,059,052.
WE COULD HAVE ALSO EVALUATED THE
COMBINATION ON THE CALCULATOR
BY PRESSING 44, MATH,
LEFT ARROW TO PROBABILITY,
OPTION 3, THEN 6, ENTER.
NOTICE HOW IT’S THE SAME VALUE.
 
SO THE PROBABILITY
IS 1/7,059,052.
 
NOW PROBABILITY COULD BE
EXPRESSED
AS A FRACTION, A DECIMAL,
OR A PERCENTAGE,
SO LET’S GO AHEAD
AND EXPRESS THIS AS A DECIMAL
AND A PERCENTAGE.
TO DO THIS, REMEMBER
THE FRACTION BAR MEANS DIVISION,
SO TO CONVERT THIS TO A DECIMAL
WE’LL HAVE 1
DIVIDED BY 7,059,052.
NOTICE HOW IT’S
A VERY SMALL NUMBER.
IT’S BEING EXPRESSED
USING SCIENTIFIC NOTATION.
SO TO WRITE THIS AS A DECIMAL
WE’D HAVE TO MOVE THE DECIMAL
TO THE LEFT 7 TIMES
SINCE THESE MEANS x 10
TO THE -7.
SO IF WE MOVE IT ONCE
WE’LL HAVE TO ADD 6 EXTRA 0’s
TO THE RIGHT
OF THE DECIMAL POINT.
SO WE’D HAVE 0.0000001417.
 
SO 0.1234561417.
EXPRESS THIS AS A PERCENTAGE
WE WOULD MULTIPLY BY 100
WHICH IS THE SAME AS MOVING
THE DECIMAL POINT
TO THE RIGHT TWO PLACES.
SO WE’D HAVE 0.00001417%.
SO YOU CAN SEE THE PROBABILITY
OF WINNING WITH ONE TICKET
IS VERY, VERY SMALL.
NOW FOR THE SECOND EXAMPLE,
TO DETERMINE THE PROBABILITY
OF WINNING WITH 2,000 TICKETS,
INSTEAD OF 1 FAVORABLE WAY WE’LL
NOW HAVE 2,000 FAVORABLE WAYS,
SO THE PROBABILITY WOULD BE
2,000 DIVIDED BY 44 CHOOSE 6
WHICH WE ALREADY KNOW
IS 7,059,052.
NOW WHEN EXPRESSING PROBABILITY
AS A FRACTION
WE SHOULD SIMPLIFY THIS,
AND SINCE THEY’RE BOTH EVEN
WE KNOW THEY’RE DIVISIBLE BY 2.
BUT SINCE THE LAST 3 DIGITS ARE
ALSO DIVISIBLE BY 4,
BOTH DIVISIBLE BY 4 AS WELL,
SO LET’S GO AHEAD AND DIVIDE
THE TOP AND BOTTOM BY 4,
SO IT WOULD GIVE US
500 DIVIDED BY 1,764,763.
AND THIS ACTUALLY DOES NOT
SIMPLIFY ANY FURTHER.
SO THE PROBABILITY OF WINNING
WITH 2,000 TICKETS
IS 500 OUT OF 1,764,763.
BUT AGAIN,
LET’S GO AHEAD AND CONVERT THIS
TO A DECIMAL AND A PERCENTAGE,
SO NOW WE’LL HAVE
500 DIVIDED BY 1,764,763.
AGAIN,
WE HAVE SCIENTIFIC NOTATION,
SO WE’LL HAVE TO MOVE
THE DECIMAL POINT
TO THE LEFT FOUR PLACES.
SO WE’LL HAVE TO ADD 3 0’s.
SO WE’LL HAVE 0.0002833,
OR APPROXIMATELY.
 
WHICH MEANS AS A PERCENTAGE
MULTIPLYING BY 100
WE WOULD HAVE APPROXIMATELY
0.02833%.
AGAIN,
STILL A VERY SMALL PERCENTAGE
OF WINNING THE LOTTERY WITH
2,000 TICKETS.
I HOPE YOU FOUND
THIS EXPLANATION HELPFUL.
THANK YOU FOR WATCHING.
 

1 thought on “Ex 1: Probability Using Combinations (Lottery)”

  1. How do you explain that some people usually (not regularly) win? They only buy a dozen tickets! Is that mere chance or math ingenuity?

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