For example, suppose you have a standard deck of 52 playing cards, and you want to find the expected value, over time, of a single card that you select at random. You need to list all possible outcomes, which are: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, in each of four different suits.
In the example of the playing cards, traditional values are Ace = 1, face cards all equal 10, and all other cards have a value equal to the number shown on the card. Assign those values for this example.
For example, with a fair coin, the probability of flipping a “Head” is 1/2, because there is one Head, divided by a total of two possible outcomes (Heads or Tails). In the example with the playing cards, there are 52 cards in the deck, so each individual card has a probability of 1/52. However, recognize that there are four different suits, and there are, for example, multiple ways to draw a value of 10. It may help to make a table of probabilities, as follows: 1 = 4/52 2 = 4/52 3 = 4/52 4 = 4/52 5 = 4/52 6 = 4/52 7 = 4/52 8 = 4/52 9 = 4/52 10 = 16/52 Check that the sum of all your probabilities adds up to a total of 1. Since your list of outcomes should represent all the possibilities, the sum of probabilities should equal 1.
For the playing card example, use the table of probabilities that you just created. Multiply the value of each card times its respective probability. These calculations will look like this: 1∗452=452{\displaystyle 1*{\frac {4}{52}}={\frac {4}{52}}} 2∗452=852{\displaystyle 2*{\frac {4}{52}}={\frac {8}{52}}} 3∗452=1252{\displaystyle 3*{\frac {4}{52}}={\frac {12}{52}}} 4∗452=1652{\displaystyle 4*{\frac {4}{52}}={\frac {16}{52}}} 5∗452=2052{\displaystyle 5*{\frac {4}{52}}={\frac {20}{52}}} 6∗452=2452{\displaystyle 6*{\frac {4}{52}}={\frac {24}{52}}} 7∗452=2852{\displaystyle 7*{\frac {4}{52}}={\frac {28}{52}}} 8∗452=3252{\displaystyle 8*{\frac {4}{52}}={\frac {32}{52}}} 9∗452=3652{\displaystyle 9*{\frac {4}{52}}={\frac {36}{52}}} 10∗1652=16052{\displaystyle 10*{\frac {16}{52}}={\frac {160}{52}}}
For the example of the playing cards, the expected value is the sum of the ten separate products. This result will be: EV=4+8+12+16+20+24+28+32+36+16052{\displaystyle {\text{EV}}={\frac {4+8+12+16+20+24+28+32+36+160}{52}}} EV=34052{\displaystyle {\text{EV}}={\frac {340}{52}}} EV=6. 538{\displaystyle {\text{EV}}=6. 538}
For example, when drawing a playing card from a standard deck, on one specific draw, the likelihood of drawing a 2 is equal to the likelihood of drawing a 6 or 7 or 8 or any other numbered card. Over many many draws, the theoretical value to expect is 6. 538. Obviously, there is no “6. 538” card in the deck. But if you were gambling, you would expect to draw a card higher than 6 more often than not.
Suppose, for this example, that you can define 4 distinct results for your investment. These results are: 1. Earn an amount equal to your investment 2. Earn back half your investment 3. Neither gain nor lose 4. Lose your entire investment
In the investment model, for simplicity, assume you invest $1. The assigned value of each outcome will be positive if you expect to earn money and negative if you expect to lose. In this problem, the four possible outcomes therefore have the following values, relative to the $1 investment: 1. Earn an amount equal to your investment = +1 2. Earn back half your investment = +0. 5 3. Neither gain nor lose = 0 4. Lose your entire investment = -1
For this example, assume that the probability of each of the four outcomes is equal, at 25%.
For the model investment situation, these calculations would look like this: 1. Earn an amount equal to your investment = +1 * 25% = 0. 25 2. Earn back half your investment = +0. 5 * 25% = 0. 125 3. Neither gain nor lose = 0 * 25% = 0 4. Lose your entire investment = -1 * 25% = -0. 25
The EV, for the stock investment model, is as follows: EV=0. 25+0. 125+0−0. 25=0. 125{\displaystyle {\text{EV}}=0. 25+0. 125+0-0. 25=0. 125}
For the investment model, a positive EV suggests that over time, you will earn money on your investments. Specifically, based on an investment of $1, you can expect to earn 12. 5 cents, or 12. 5% of your investment. Earning 12. 5 cents does not sound impressive. However, applying the calculation to large numbers suggests, for example, that an investment of $1,000,000 would earn $125,000.
1 = -$10 2 = -$10 3 = -$10 4 = -$10 5 = $20 win - $10 bet = +$10 net value 6 = $30 win - $10 bet = +$20 net value
1 = -$10 * 0. 167 = -1. 67 2 = -$10 * 0. 167 = -1. 67 3 = -$10 * 0. 167 = -1. 67 4 = -$10 * 0. 167 = -1. 67 5 = $20 win - $10 bet = +$10 net value * 0. 167 = +1. 67 6 = $30 win - $10 bet = +$20 net value * 0. 167 = +3. 34
EV=−1. 67−1. 67−1. 67−1. 67+1. 67+3. 34=−1. 67{\displaystyle {\text{EV}}=-1. 67-1. 67-1. 67-1. 67+1. 67+3. 34=-1. 67}
If you play the game once, you might win $30 (net +$20). If you play a second time, you could even win again, for a total of $60 (net +$40). However, that luck is not going to continue if you keep playing. If you play 100 times, in the end you are likely to be down approximately $167.
