7C                 The Law of Averages

    if you toss a fair coin 100 times, heads and tails will both come up close to half the time.

Consider events A and B, each with its own value and probability  P(A),  P(B).
The expected value is

expected value = (value of A) * P(A)  + (value of B) * P(B)
 

Bernoulli: "The value of your expectations always signifies something in the middle between the best we can hope for and the worst we can fear."
 

• (3) You are given 5 to 1 odds against tossing three heads in 3 tosses of a fair coin, meaning you win  $5, if you succeed and you lose  $1 if you fail. Find the expected value. Do you expect to lose in 1 game, in 100 games?

 

 
 
 

Solution:
 
 

game outcomes and their values

game outcomes	value
HHH	+ 5
HHT	-1
HTH	-1
HTT	-1
THH	-1
THT	-1
TTH	-1
TTT	-1

expected value = 5 * P(HHH) - 1 * P(HHT) -  1 * P(HTH)  -  1 * P(HTT)  -  1 * P(THH)  -  1 * P(THT)  -  1 * P(TTH)  -  1 * P(TTT)
                       = 5 * 1/8 - 1 * 7/8
                       = - 2/8
                       = - 1/4
the outcome of 1 game cannot be predicted; but if you play, say, 100 games you can expect to lose about a quarter per game, i.e. over a hundred games you would lose about  $25.
 

• (6) You are given 10 to 1 odds against rolling a double number (1 + 1, 2 + 2, ...) with a roll of two fair dice, i.e. you win  $10 if you succeed and you lose  $1 if you fail. Find the expected value. Do you expect to lose in 1 game, in 100 games.

• (7) An insurance policy sells for $1000. Based on past data,

an average of  1 in 100 policy holders will file a  $20,000 claim,
an average of  1 in 200 policy holders will file a  $50,000 claim,
an average of  1 in 500 policy holders will file a  $100,000 claim,
if the company sells 100,000 policies, wehat is the profit or loss?

Solution:

expected value = 1000 * 1 - 20,000 * 1/100 - 50,000 * 1/200 - 100,000 * 1/500 = 350

expected gain per policy.
expected profit from 100,000 policies sold: $35,000,000

• (9) Suppose you arrive at a bus stop randomly, so all arrival times are equally likely. The bus arrives regularly every 30 minutes without delay ( say on the hour and on the haldf hour). What is the expected value of your waiting time? Explain.
• (12) Given the table of odds for winning a prize in the Powerball lottery. ($100 and $7 can be won in two ways each) The jackpot is $100 million. Find the expected value of the winnings. Each ticket is  $1. How much can you expect to win or lose if you spend  $365 per year on the lottery?

 

Prize	Probability
$100 million	1 in 80,089,128
$100,000	1 in 1,953,393
$5000	1 in 364,042
$100	1 in 8879
$100	1 in 8466
$7	1 in 207
$7	1 in 605
$4	1 in 188
$3	1 in 74

Gambler's fallacy : mistaken belief that a streak of bad luck makes a person "due" for a streak of good luck.

• (19) Suppose you roll a die with the following rules:
• die is even, you win $1
• die is odd, you lose $1
a)    Suppose you get 45 even numbers in your first 100 rolls. How much money have you won or lost?
b)    On the second 100 rolls, your luck improves and you roll 47 even numbers.
        How much money have you won or lost over 200 rolls?
c)    Your luck continues to improve and you roll 148 even numbers in your next 300 rolls.
        How much money have you won or lost over your total of 500 rolls?
d)    How many even numbers would you have to roll in the next 100 rolls to break even? Is this likely? Explain.
e)    What were the percentages of even numbers after 100, 200, and 500 rolls?
        Explain, how this illustrates the law of averages, even while your losses increased.

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