Question

When you give $4 for a bet in a casino​ game, there is a 258/495 probability that you will lose $4 and there is a 237/495 probability that you will make a net gain of $4. ​(If you​ win, the casino gives you $ 4and you get to keep your $ 4​bet, so the net gain is $4.)

(a)- What is your expected​ value?
(b)-In the long​ run, how much do you lose for each dollar​ bet?

Answer 1

The expected value of the bet is -$2.38, and in the long run, you would expect to lose $2.38 for each dollar bet.

Step-by-step explanation:

The expected value of a bet can be calculated by multiplying the probability of each outcome by its corresponding value, and summing up the results.

In this case, the expected value is calculated as follows:

Expected value = (Probability of losing) * (Value of losing) + (Probability of winning) * (Value of winning)

Expected value = (258/495) * (-4) + (237/495) * 4 = -2.38

Therefore, the expected value of the bet is -$2.38.

In the long run, for each dollar bet, you would expect to lose $2.38 on average.

Answer 2

a) - $0.169

b) - $0.04242

Step-by-step explanation:

Data provided in the question:

Bet amount = $4

Probability of losing $4 =
`(258)/(495)`

Probability of gaining $4 =
`(237)/(495)`

a) The expected value

= Winning amount × winning probability - Losing amount × Losing probability

= $4 ×
`(237)/(495)` - $4 ×
`(258)/(495)`

= $4 ×
`(237-258)/(495)`

= - $0.169 (Negative sign means loss)

b) The amount lost for each dollar =
`\frac{\textup{Expected Value}}{\textup{Bet amount}}`

=
`\frac{\textup{-0.169}}{\$\textup{4}}`

= - $0.04242