Suppose you roll a die with a friend, with the following rules: For every even number you roll, you win $1 from your friend; for every odd number you roll, you pay $1 to your friend.
a. What are the chances of rolling an even number on one roll of a fair die? An odd number?
b. Suppose that on the first 100 rolls, you roll 45 even numbers. How much money have you won or lost?
c. Suppose that 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?
d. Suppose that over the next 300 rolls, your luck again improves and you roll 148 even numbers. How much money have you won or lost over 500 rolls?
e. What was the percentage of even numbers after 100, 200, and 500 rolls? Explain why this game illustrates the gambler's fallacy.
f. How many even numbers would you have to roll in the next 100 rolls to break even?