In statistics and probability, the formula for expected value is E(X) = summation of X * P(X), or the sum of all gains multiplied by their individual probabilities. The expected value is comprised on two components: how much you can expect to gain, and how much you can expect to lose. It's the sum of both your expected losses and expected gains in any situation from gambling on the stock market to buying a lottery ticket.
Draw or sketch a two-row probability chart. The probability chart should be two rows wide and two columns deep. Label the first row "Gain" and the second row "Probability." Label the first column "Win" and the second column "Lose."
Work out how much money you could win or lose. For example, you buy a lottery ticket for $10. You have one chance of winning $10,000. Your potential loss is the cost of the ticket: $10. Your potential gain is $10,000 less the cost of the $10 ticket ($10,000 -- $10) = $9,990.
Place the values from Step 2 in the relevant spaces on the probability chart---$9,990 should go at the top left and --$10 should go at the top right. Note the negative sign on the $10 to denote a loss.
Write your odds of winning on the bottom row. If the lottery has sold 1,000 tickets, your odds of winning are 1 in 1,000, or 1/1000. Place this at the bottom left of the probability chart.
Write your odds of losing on the bottom row. If your odds of winning are 1/1000, then your odds of losing are 999 out of 1,000, or 999/1000.
Multiply the figure at the top of each column by the figure at the bottom of each column. In the above example, the calculations for the expected values are:
--$10 * 999/1000 = --9.99 $9,990 * 1/1000 = 9.99
Add all of the values together to compute the expected value. In the above example, --9.99 + 9.99 = 0, so your expected value from purchasing one ticket in the lottery is zero.
For more than one item, just extend the probability chart to the right, adding more columns for more items. Remember to add all of the probabilities multiplied by their gains.
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