In the game of roulette, a player can place a $4 bet on the number 13 and have a [tex]$\frac{1}{38}$[/tex] probability of winning. If the metal ball lands on 13, the player gets to keep the $4 paid to play the game and the player is awarded an additional $140. Otherwise, the player is awarded nothing and the casino takes the player's $4. 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?
The expected value is $ [ ]
(Round to the nearest cent as needed.)
Answer (1)
Calculate the expected value: E = ( P ( winning ) × amount won ) + ( P ( losing ) × amount lost ) .
Substitute the given values: E = ( 38 1 × 144 ) + ( 38 37 × − 4 ) .
Simplify the expression: E = 38 − 4 ≈ − 0.10526 .
Round to the nearest cent: − $0.11 .
Explanation
Understand the problem and provided data Let's analyze the game of roulette. A player bets 4 o n t h e n u mb er 13. T h e p ro babi l i t yo f w innin g i s \frac{1}{38}$. If the player wins, they get to keep their $4 bet and are awarded an additional $140, for a total of $144. If the player loses, they lose their $4 bet. We want to find the expected value of the game to the player and the expected loss after playing the game 1000 times.
Define the expected value formula To calculate the expected value of the game, we need to consider the probability of winning and the amount won, as well as the probability of losing and the amount lost. The formula for expected value is: E = ( P ( winning ) × amount won ) + ( P ( losing ) × amount lost )
Identify probabilities and amounts In this case:
P ( winning ) = 38 1
Amount won = 144 − P(\text{losing}) = \frac{37}{38}$ - Amount lost = -$4
Calculate the expected value Now, we can plug these values into the formula: E = ( 38 1 × 144 ) + ( 38 37 × − 4 ) E = 38 144 − 38 148 E = 38 144 − 148 E = 38 − 4 E = − 19 2 E ≈ − 0.10526
Round the expected value Rounding the expected value to the nearest cent, we get: E ≈ − $0.11
Calculate the expected loss after 1000 plays If the player plays the game 1000 times, the expected loss would be: Expected loss = 1000 × E Expected loss = 1000 × − $0.10526 Expected loss = − $105.26
State the final answer Therefore, the expected value of the game to the player is approximately − $0.11 , and if the player played the game 1000 times, they would expect to lose approximately $105.26 .
Examples
Roulette, while a game of chance, can be analyzed using expected value to understand potential long-term outcomes. For example, casinos use expected value calculations to ensure profitability over many plays. Similarly, understanding expected value can help in making informed decisions in other games of chance or investment scenarios, where assessing potential gains and losses is crucial. By calculating the expected value, one can determine whether participating in the game is advantageous or disadvantageous in the long run.
