As a project for math class, two students devised a game in which 3 black marbles and 2 red marbles are put into a bag. First the players must decide who is playing black marbles and who is playing red marbles. Then each player takes a turn drawing a marble, noting the color, replacing the marble in the bag, and then drawing a second marble and noting the color before returning it to the bag. The point scheme for the game is detailed in the table.
| | |
| :-------------------- | :-------------------- |
| Point Values for Marble Game | |
| Black Marble Points | Red Marble Points |
| Both black: +2 points | Both red: +4 points |
| Different colors: -1 point | Different colors: -1 points |
| Both red: 0 points | Both black: 0 points |
If Seth is challenged to a game by a classmate, which statement is correct in all aspects in helping him make the correct choice?
A. Since [tex]E ( red )= E ( black )[/tex], it is a fair game, so it doesn't matter which color Seth chooses.
B. Since [tex]E( black )=0.24[/tex] and [tex]E ( red )=0.16[/tex], Seth should choose to play black marbles.
C. E (red) will be twice that of E (black), so Seth should choose to play red marbles.
D. Both options will lose points because there are two ways to lose points and only one way to gain points. He should choose neither color.
Answer (2)
Seth should choose to play black marbles as the expected value for black marbles is higher (0.24) compared to red marbles (0.16). The correct answer is option B. Therefore, opting for black marbles maximizes his potential points.
;
Calculate the probability of drawing two black marbles: P ( " B o t h Bl a c k " ) = 5 3 × 5 3 = 0.36 .
Calculate the probability of drawing two red marbles: P ( " B o t h R e d " ) = 5 2 × 5 2 = 0.16 .
Calculate the expected values: E ( " Bl a c k " ) = 0.24 and E ( " R e d " ) = 0.16 .
Since E("Red")"> E ( " Bl a c k " ) > E ( " R e d " ) , Seth should choose to play black marbles: E ( " Bl a c k " ) = 0.24 .
Explanation
Analyze the game and define probabilities Let's analyze the game to determine the best strategy for Seth. We need to calculate the expected value of choosing black marbles and the expected value of choosing red marbles. The probabilities and point values will help us determine which choice gives Seth a higher expected value.
Calculate the probabilities of each outcome First, let's calculate the probabilities for each outcome. There are 3 black marbles and 2 red marbles, making a total of 5 marbles. When drawing with replacement, the probabilities remain constant for each draw.
Probability of drawing two black marbles: P ( " B o t h Bl a c k " ) = 5 3 × 5 3 = 25 9 = 0.36
Probability of drawing two red marbles: P ( " B o t h R e d " ) = 5 2 × 5 2 = 25 4 = 0.16
Probability of drawing different colored marbles: P ( " D i ff ere n tC o l ors " ) = 1 − P ( " B o t h Bl a c k " ) − P ( " B o t h R e d " ) = 1 − 0.36 − 0.16 = 0.48
Calculate the expected value for black marbles Now, let's calculate the expected value for choosing black marbles. The point values are +2 for both black, -1 for different colors, and 0 for both red.
E ( " Bl a c k " ) = ( + 2 ) × P ( " B o t h Bl a c k " ) + ( − 1 ) × P ( " D i ff ere n tC o l ors " ) + ( 0 ) × P ( " B o t h R e d " ) E ( " Bl a c k " ) = ( 2 × 0.36 ) + ( − 1 × 0.48 ) + ( 0 × 0.16 ) = 0.72 − 0.48 + 0 = 0.24
Calculate the expected value for red marbles Next, let's calculate the expected value for choosing red marbles. The point values are +4 for both red, -1 for different colors, and 0 for both black.
E ( " R e d " ) = ( 0 ) × P ( " B o t h Bl a c k " ) + ( − 1 ) × P ( " D i ff ere n tC o l ors " ) + ( + 4 ) × P ( " B o t h R e d " ) E ( " R e d " ) = ( 0 × 0.36 ) + ( − 1 × 0.48 ) + ( 4 × 0.16 ) = 0 − 0.48 + 0.64 = 0.16
Compare expected values and determine the best choice Comparing the expected values, we have:
E ( " Bl a c k " ) = 0.24 E ( " R e d " ) = 0.16
Since E("Red")"> E ( " Bl a c k " ) > E ( " R e d " ) , Seth should choose to play black marbles to maximize his expected value.
State the final answer Therefore, the correct statement is: Since E ( " Bl a c k " ) = 0.24 and E ( " R e d " ) = 0.16 , Seth should choose to play black marbles.
Examples
This type of probability calculation is useful in many real-world scenarios, such as in business when deciding which marketing strategy to use. By calculating the expected value of each strategy, a business can determine which strategy is most likely to be successful and generate the most profit. It's also applicable in games of chance, helping players make informed decisions to maximize their potential winnings.
