- WELCOME TO A LESSON ON EXPECTED VALUE.
IN THIS LESSON WE WILL DEFINE EXPECTED VALUE
AND ALSO DETERMINE THE EXPECTED VALUE OF AN EVENT.
EXPECTED VALUE IS AN EXTREMELY USEFUL PROBABILITY CONCEPT
TO LEARN.
IT HAS MANY APPLICATIONS RANGING FROM INSURANCE POLICES
TO MAKING FINANCIAL DECISIONS.
IT IS ALSO ONE THING THAT THE CASINOS AND GOVERNMENT AGENCIES
THAT RUN GAMBLING OPERATIONS AND LOTTERIES
HOPE MOST PEOPLE WON'T LEARN.
EXPECTED VALUE IS THE AVERAGE GAIN OR LOSS OF AN EVENT
IF THE PROCEDURE IS REPEATED MANY TIMES.
WE CAN COMPUTE THE EXPECTED VALUE
BY MULTIPLYING EACH OUTCOME BY THE PROBABILITY OF THAT OUTCOME,
THEN ADDING UP THE PRODUCTS.
SO LETS TAKE A LOOK AT AN EXAMPLE.
YOU PURCHASE ONE RAFFLE TICKET FOR A SCHOOL FUNDRAISER.
THE RAFFLE TICKET COSTS $5.00.
THE SCHOOL IS SELLING 10,000 TICKETS.
ONE TICKET WILL BE RANDOMLY DRAWN
AND THE WINNER WILL RECEIVE $10,000.
ASSUME ALL TICKETS ARE SOLD.
COMPUTE THE EXPECTED VALUE.
SO AGAIN, THE EXPECTED VALUE WOULD BE THE EXPECTED OUTCOME
IF YOU PLAYED THIS LOTTERY OVER AND OVER AND OVER AGAIN.
SO THE FIRST THING WE NEED TO RECOGNIZE
IS THAT THERE ARE 10,000 TICKETS
AND YOU ONLY PURCHASE ONE
THEREFORE THE PROBABILITY OF WINNING IS 1/10,000th
WHICH MEANS THE PROBABILITY OF LOSING WOULD BE ONE
MINUS THE PROBABILITY OF WINNING OR 9,999 DIVIDED BY 10,000.
NEXT, IF YOU WIN YOU DON'T REALLY WIN $10,000,
YOU'D ACTUALLY HAVE TO SUBTRACT THE COST OF THE TICKET.
SO IF YOU WIN YOU ACTUALLY WIN $9,995
AND OF COURSE IF YOU LOSE THEN YOU LOSE YOUR $5.00
FROM PURCHASING THE TICKET.
THIS IS THE INFORMATION WE NEED
TO FIND THE EXPECTED VALUE FOR THIS EVENT.
EXPECTED VALUE IS GOING TO BE EQUAL TO THE OUTCOME OF WINNING
x THE PROBABILITY OF WINNING + THE OUTCOME OF LOSING
x THE PROBABILITY OF LOSING.
SO AGAIN, IF WE WIN, WE WIN $9,995.
THE PROBABILITY OF THIS OCCURRING IS 1/10,000th.
IF WE LOSE, WE LOSE $5.00.
THE PROBABILITY OF THIS OCCURRING
IS THIS PROBABILITY HERE.
SO WE FIND THIS PRODUCT, THIS PRODUCT
AND THEN FIND THE SUM TO FIND THE EXPECTED VALUE.
SO $9,995 x 1/10,000th = $.9995
AND THEN -5 x THIS PROBABILITY = $-4.9995.
FIND THE SUM HERE.
THE RESULT IS -4 OR -$4.00.
THIS IS OUR EXPECTED VALUE FOR THIS EVENT.
WHICH MEANS IF YOU WERE TO PLAY THIS LOTTERY
TIME AFTER TIME AFTER TIME
THE EXPECTED RESULT WOULD BE
THAT YOU WOULD LOSE $4.00 EACH TIME.
LET'S TAKE A LOOK AT A SECOND EXAMPLE.
A COMPANY ESTIMATES THAT .2% OF THEIR PRODUCTS WILL FAIL
AFTER THE ORIGINAL WARRANTY PERIOD
BUT WITHIN TWO YEARS OF THE PURCHASE
WITH THE REPLACEMENT COST OF $250.
IF THEY OFFER A 2-YEAR EXTENDED WARRANTY FOR $28,
WHAT IS THE COMPANY'S EXPECTED VALUE OF EACH WARRANTY.
SO WE'RE THINKING OF THIS AS IF WE ARE THE COMPANY
WHICH MEANS THE PROBABILITY OF HAVING TO REPLACE THE PRODUCT
WOULD BE .2% OR AS A DECIMAL .002.
WHICH MEANS THE PROBABILITY OF NOT HAVING TO REPLACE A PRODUCT
WOULD BE 1 - .002 OR .998 OR 99.8%.
NEXT, THE COST OF REPLACEMENT UNDER WARRANTY
WOULD NOT BE $250.
IT WOULD BE THE COST OF THE WARRANTY
MINUS THE REPLACEMENT COST.
SO NOTICE HOW THE TRUE REPLACEMENT COST FOR THE COMPANY
IS $222,
AND AGAIN THIS IS NEGATIVE
BECAUSE IT'S COSTING THE COMPANY MONEY.
IF THEY'RE GAINING MONEY IT WOULD BE POSITIVE.
WHICH MEANS THE REVENUE FROM NON-REPLACEMENT
WOULD BE THE $28 CHARGED FOR EACH WARRANTY.
SO ONE THING WE DO HAVE TO BE CAREFUL ABOUT HERE
IS WATCHING THE SIGNS WHEN WE'RE GATHERING THIS INFORMATION.
THIS IS THE INFORMATION WE NEED
IN ORDER TO FIND THE EXPECTED VALUE FOR THE COMPANY.
THE EXPECTED VALUE IS GOING TO BE EQUAL
TO THE REPLACEMENT COST OF -$222
x THE PROBABILITY OF REPLACING THE ITEM,
WHICH AS A DECIMAL IS .002
+ THE EXTRA REVENUE FROM NOT HAVING TO REPLACE THE ITEM
WHICH IS $28,
THE COST OF THE WARRANTY
x THE PROBABILITY OF NOT HAVING TO REPLACE THE PRODUCT,
WHICH AS A DECIMAL WOULD BE .998.
SO IF WE FIND THE SUM OF THIS PRODUCT
WE HAVE THE EXPECTED VALUE FOR THE COMPANY.
SO THIS FIRST PRODUCT COMES OUT TO -$.444.
THE SECOND PRODUCT COMES OUT TO $27.944.
FINDING THE SUM HERE
NOTICE HOW THE RESULT OF THE EXPECTED VALUE IS $27.50.
WHICH MEANS AS THEY SELL MORE AND MORE WARRANTIES
THE EXPECTED VALUE OR ACTUAL REVENUE
WOULD BE $27 PER WARRANTY SOLD.
I HOPE YOU FOUND THIS LESSON HELPFUL.