We are told that the cost of a bet is $6. The bettor wins with a probability of 1/38, and the yield is $210 (they get their $6 entry back). The bettor loses with a probability 37/38 and loses their $6 entry.

We have a probability distribution:

`{[x,-6,210],[P(x),37/38,1/38]}`

Note that...

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We are told that the cost of a bet is $6. The bettor wins with a probability of 1/38, and the yield is $210 (they get their $6 entry back). The bettor loses with a probability 37/38 and loses their $6 entry.

We have a probability distribution:

`{[x,-6,210],[P(x),37/38,1/38]}`

Note that the probabilities sum to 1 and that all possibilities have been covered (as the bettor either wins or loses at each play).

We can calculate the mean and standard deviation of a probability distribution. We essentially treat it as a weighted mean of a distribution.

`mu=-6(1/38)+210(37/38)~~-.3158`

The key is to understand that the expected value (or expectation) of the game is the mean of the underlying probability distribution.

Thus, `E(x)=mu~~-0.3158`

Obviously, a player will never lose 32 cents at one game. Either the player loses $6 or wins $210. But over time, the average loss (in this case) is between 31 and 32 cents. If the player played 100 times, they should expect to lose about $31.58.

In a calculator that has tables, you can enter the possible wins/losses in one column and their corresponding probabilities in another column and compute the two-variable statistics on those columns. In a calculator without tables, you can enter the numerators as the frequencies (assuming the denominators of the probabilities are the same) and run two-variable statistics.