CALCULATING EXPECTED VALUES

Consider the following scenario involving a well-balanced coin: whenever a head appears, you earn $1 and whenever a tail appears, you earn $1 dollar. If you toss the coin 100 times, how much money do you expect to earn? Since you will earn $1 regardless of the outcome, the expected value (in fact, the guaranteed value) is 100.

Now consider this scenario: whenever a head appears, you earn $1 and whenever a tail appears, you earn 0 dollars. If you toss the coin 100 times, how much money do you expect to earn? You probably determined the value 50 (which is the correct answer) by making a quick mental calculation. The more formal derivation of the value of E (the expected earning) is here:

E = 100 *[1 * 0.5 + 0 * 0.5] = 100 * 0.5 = 50

The quantity 1 * 0.5 + 0 * 0.5 is the amount of money you expected to collect during each coin toss (half the time you collect $1 and half the time you collect 0 dollars), and multiplying this number by 100 is the expected amount...