Monica A.

asked • 07/12/19

statistics question

Lottery: In the New York State Numbers Lottery, you pay $1 and can bet that the sum of the numbers that come up is 13. The probability of winning is 0.075, and if you win, you win $6.50, which is a profit of $5.50. If you lose, you lose $1. What is the expected value of your profit? Is it an expected gain or an expected loss?

Colin L.

Hi Monica, Let's see if I can break this question down into simpler terms. It seems that what it's describing is a system of outcomes - there's a 0.075 chance of winning and gaining $5.50, but if you don't win, you lose $1. Now, because there's a 0.075 chance that you win $6.50 (and that results in a profit of $5.50 when your original $1 is subtracted) we multiply this probability of 0.075 by this net gain, 5.50, to find .4125 (ignoring the units for now as we don't need them yet, this is only one event). However, in this situation there's another possibility of losing $1, which has an apparent probability of 1-0.075 = 0.925 of happening (if you don't win, you lose). Because this is another event, we multiply the probability by its respective net gain of -1 to find a value of -0.925 (think of this in terms of large numbers, if you multiply the average probability of .925 by 1000 trials in this lottery, you'd find an average loss of $925 per $1000 spent, and an average gain of about $413 from the 5.50 x 75 times you won.) Because we're looking for the average outcome, though, we take this value of -.925 and add it to 0.4125 to find an average gain of about 0.4125 - 0.925 = -0.51 dollars, or, in more reasonable terms, an expected loss of 51 cents per dollar you spend in this lottery.
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07/12/19

1 Expert Answer

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Rich G. answered • 07/12/19

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To find the expected gain/loss, multiply the probability by the outcome and add them together.


Since there's a 0.075 chance of a profit of $5.50, there's a 0.925 chance of a loss of $1:


0.075 ($5.50) - .925($1) = $0.4125 - $0.925 = -$0.5125


So the expected value is - $0.5125. In other words, every time someone plays the expected loss is about 51 cents.

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