Question

In the game of​ roulette, a player can place a ​$7 bet on the number 5 and have a startfraction 1 over 38 endfraction probability of winning. if the metal ball lands on 5, the player gets to keep the ​$7 paid to play the game and the player is awarded an additional ​$245. ​ otherwise, the player is awarded nothing and the casino takes the​ player's ​$7. what is the expected value of the game to the​ player? if you played the game 1000​ times, how much would you expect to​ lose?

Answer 1

The expected value of one game is approximately -$0.3684. If you played the game 1000 times, you would expect to lose around -$368.40 on average. The negative sign indicates that, on average, the player is expected to lose money over multiple plays.

Explanation:

To calculate the expected value (EV) of the game for the player, you can use the following formula:

`\[ EV = (P_{\text{win}} * \text{Winnings}) + (P_{\text{lose}} * \text{Loss}) \]`

Where:

`\( P_{\text{win}} \)` is the probability of winning.

`\( P_{\text{lose}} \)` is the probability of losing.

`\text{Winnings}` is the amount won when the player wins.

`\text{Loss}` is the amount lost when the player loses.

In this case:

`\( P_{\text{win}} = (1)/(38) \)` (probability of winning)

`\( P_{\text{lose}} = 1 - P_{\text{win}} = (37)/(38) \)`(probability of losing)

\text{Winnings} = $245

`\text{Loss} = \$7`

Now, plug in these values into the formula:

`\[ EV = \left( (1)/(38) * 245 \right) + \left( (37)/(38) * (-7) \right) \]`

`\[ EV = \left( (245)/(38) \right) - \left( (259)/(38) \right) \]`

`\[ EV = -(14)/(38) \]`

`\[ EV \approx -0.3684 \]`

So, the expected value of one game is approximately (-$0.3684).

If you played the game 1000 times, you can estimate the expected total loss by multiplying the expected value by the number of times you played:

`\[ \text{Total Expected Loss} = EV * \text{Number of Games} \]`

`\[ \text{Total Expected Loss} = -0.3684 * 1000 \]`

`\[ \text{Total Expected Loss} = -\$368.40 \]`

Therefore, if you played the game 1000 times, you would expect to lose approximately $368.40 on average.