Problem 1031

Find the expected payback for a game in which you bet 20 on any number from 1 to 100. If your number comes up, you get2000.




Let x, the random variable, represent the possible amounts of payback, where payback is the amount won less the cost.

The payback for winning is 2000 -20  = 1980.

The payback for losing is 0 -20 = - 20.  Now consider the probabilities of winning and losing. The probability of winning is the probability of drawing the picked number form the possible 100 numbers. The probability of losing is the probability of drawing any number but the picked one. <table> <tbody> <tr> <td width="145"> </td> <td width="145"> </td> <td width="145">Winning</td> <td width="145">Losing</td> </tr> <tr> <td width="145">x</td> <td width="145">(payback)</td> <td width="145">1980 - 20</td> </tr> <tr> <td width="145">P(x)</td> <td width="145">(probability)</td> <td width="145">0.0100</td> <td width="145">0.9900</td> </tr> </tbody> </table> The expected payback is calculated using the expected values formula, E(x) = x<sub>1</sub>p<sub>1</sub> + …..+x<sub>n</sub>p<sub>n</sub>.  Substitute the values of x and p into the expected value formula and calculate.  E(x) = (1980)-(0.0100 ) +(-20)*(0.9900) =0.00.

Therefore , the expected payback is $0.00.

Note that a game with an expected value of 0 is called a fair game.



Leave a Reply

Your email address will not be published. Required fields are marked *