In the game of roulette, a player can place a $6 bet on the number 10 and have a 1 / 38 probability of winning. If the metal ball lands on 10, the player gets to keep the $6paid to play the game and the player is awarded an additional $210. Otherwise, the player is awarded nothing, and the casino takes the player's $6. Find the expected value E(x) to the player for one play of the game. If x is the gain to a player in a game of chance, then E(x) is usually negative. This value gives the average amount per game the player can expect to lose. What is the expected value?

The expected value of a game where losing costs $6 with a probability of 37/38 and winning yields $210 with a probability of 1/38 is approximately -$0.32.

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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...

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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.

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