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Sagot :
To determine the expected value of the number of points Harlene gets for one roll of two number cubes, let's break down the problem step-by-step.
First, we should understand the possible outcomes of rolling two six-sided dice and their corresponding probabilities.
### Possible Outcomes:
1. Sum of 8:
- (2, 6)
- (3, 5)
- (4, 4)
- (5, 3)
- (6, 2)
These are 5 outcomes that result in a sum of 8.
2. Sum of 12:
- (6, 6)
This is the only outcome that results in a sum of 12.
### Total Number of Outcomes:
Since each die has 6 faces, the total number of outcomes when rolling two dice is:
[tex]\[ 6 \times 6 = 36 \][/tex]
### Probabilities:
1. Probability of the sum being 8:
[tex]\[ \text{Number of outcomes resulting in a sum of 8} = 5 \][/tex]
[tex]\[ \text{Probability of sum being 8} = \frac{5}{36} \approx 0.1389 \][/tex]
2. Probability of the sum being 12:
[tex]\[ \text{Number of outcomes resulting in a sum of 12} = 1 \][/tex]
[tex]\[ \text{Probability of sum being 12} = \frac{1}{36} \approx 0.0278 \][/tex]
3. Probability of other outcomes (not 8 or 12):
[tex]\[ \text{Probability of sum NOT being 8 or 12} = 1 - \left(\frac{5}{36} + \frac{1}{36}\right) \][/tex]
[tex]\[ = 1 - \frac{6}{36} \][/tex]
[tex]\[ = 1 - \frac{1}{6} \][/tex]
[tex]\[ = \frac{30}{36} = \frac{5}{6} \approx 0.8333 \][/tex]
### Points System:
- Getting a sum of 8 or 12: 9 points
- Getting any other sum: -2 points
### Expected Value Calculation:
To find the expected value, multiply the points for each scenario by their respective probabilities and sum the results:
[tex]\[ E(X) = \left(\frac{5}{36} \times 9 \right) + \left(\frac{1}{36} \times 9 \right) + \left(\frac{30}{36} \times (-2)\right) \][/tex]
Let's break it down:
- For the sum of 8:
[tex]\[ \frac{5}{36} \times 9 = \frac{45}{36} = 1.25 \][/tex]
- For the sum of 12:
[tex]\[ \frac{1}{36} \times 9 = \frac{9}{36} = 0.25 \][/tex]
- For all other sums:
[tex]\[ \frac{30}{36} \times (-2) = \frac{30}{36} \times -2 = -\frac{60}{36} = -1.6667 \][/tex]
Add these contributions together for the expected value:
[tex]\[ E(X) = 1.25 + 0.25 - 1.6667 = -0.1667 \][/tex]
Thus, the expected value of the number of points for one roll is approximately [tex]\(-0.1667\)[/tex].
### Answer:
The expected value from the options provided is:
[tex]\[ -\frac{1}{6} \][/tex]
So, the correct answer is [tex]\( -\frac{1}{6} \)[/tex].
First, we should understand the possible outcomes of rolling two six-sided dice and their corresponding probabilities.
### Possible Outcomes:
1. Sum of 8:
- (2, 6)
- (3, 5)
- (4, 4)
- (5, 3)
- (6, 2)
These are 5 outcomes that result in a sum of 8.
2. Sum of 12:
- (6, 6)
This is the only outcome that results in a sum of 12.
### Total Number of Outcomes:
Since each die has 6 faces, the total number of outcomes when rolling two dice is:
[tex]\[ 6 \times 6 = 36 \][/tex]
### Probabilities:
1. Probability of the sum being 8:
[tex]\[ \text{Number of outcomes resulting in a sum of 8} = 5 \][/tex]
[tex]\[ \text{Probability of sum being 8} = \frac{5}{36} \approx 0.1389 \][/tex]
2. Probability of the sum being 12:
[tex]\[ \text{Number of outcomes resulting in a sum of 12} = 1 \][/tex]
[tex]\[ \text{Probability of sum being 12} = \frac{1}{36} \approx 0.0278 \][/tex]
3. Probability of other outcomes (not 8 or 12):
[tex]\[ \text{Probability of sum NOT being 8 or 12} = 1 - \left(\frac{5}{36} + \frac{1}{36}\right) \][/tex]
[tex]\[ = 1 - \frac{6}{36} \][/tex]
[tex]\[ = 1 - \frac{1}{6} \][/tex]
[tex]\[ = \frac{30}{36} = \frac{5}{6} \approx 0.8333 \][/tex]
### Points System:
- Getting a sum of 8 or 12: 9 points
- Getting any other sum: -2 points
### Expected Value Calculation:
To find the expected value, multiply the points for each scenario by their respective probabilities and sum the results:
[tex]\[ E(X) = \left(\frac{5}{36} \times 9 \right) + \left(\frac{1}{36} \times 9 \right) + \left(\frac{30}{36} \times (-2)\right) \][/tex]
Let's break it down:
- For the sum of 8:
[tex]\[ \frac{5}{36} \times 9 = \frac{45}{36} = 1.25 \][/tex]
- For the sum of 12:
[tex]\[ \frac{1}{36} \times 9 = \frac{9}{36} = 0.25 \][/tex]
- For all other sums:
[tex]\[ \frac{30}{36} \times (-2) = \frac{30}{36} \times -2 = -\frac{60}{36} = -1.6667 \][/tex]
Add these contributions together for the expected value:
[tex]\[ E(X) = 1.25 + 0.25 - 1.6667 = -0.1667 \][/tex]
Thus, the expected value of the number of points for one roll is approximately [tex]\(-0.1667\)[/tex].
### Answer:
The expected value from the options provided is:
[tex]\[ -\frac{1}{6} \][/tex]
So, the correct answer is [tex]\( -\frac{1}{6} \)[/tex].
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