Math / Data & Statistics

QuestionA board game involves tossing a fair, regular die (with $1,2,3,4,5$ , and 6 dots on the faces) and moving the number of spaces indicated by the number of dots. If the die is tossed many times, what is the expected value of the number of spaces to be moved? (Round your answer to one decimal place if applicable.) expected Value: $\square$
What does that mean in context? This means that with a large number of tosses, the average number of spaces moved is 3.5 . This means that the most likely dice roll will be about 3.5 . This means the number of spaces moved the next time the die is rolled will probably be around 3.5 . The average number of spaces moved is 3.5 .

Studdy Solution

STEP 1

1. The die is fair, meaning each face has an equal probability of landing face up.
2. The die has six faces numbered from 1 to 6.
3. The expected value is calculated as the average outcome over many trials.

STEP 2

1. Identify the possible outcomes and their probabilities.
2. Calculate the expected value using the formula for expected value.

STEP 3

List the possible outcomes when a die is tossed: $1, 2, 3, 4, 5, 6$ .

STEP 4

Determine the probability of each outcome. Since the die is fair, each outcome has a probability of:
$\frac{1}{6}$

STEP 5

Use the formula for expected value $E(X)$ , which is the sum of each outcome multiplied by its probability:
$E(X) = \sum_{i=1}^{6} x_i \cdot P(x_i)$
where $x_i$ are the outcomes and $P(x_i)$ is the probability of each outcome.

STEP 6

Calculate the expected value:
$E(X) = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + 3 \cdot \frac{1}{6} + 4 \cdot \frac{1}{6} + 5 \cdot \frac{1}{6} + 6 \cdot \frac{1}{6}$
$E(X) = \frac{1 + 2 + 3 + 4 + 5 + 6}{6}$
$E(X) = \frac{21}{6}$
$E(X) = 3.5$
The expected value of the number of spaces to be moved is:
$\boxed{3.5}$
In context, this means: - With a large number of tosses, the average number of spaces moved is 3.5. - The statement "This means that the most likely dice roll will be about 3.5" is incorrect because 3.5 is not an outcome of a single die roll. - The statement "This means the number of spaces moved the next time the die is rolled will probably be around 3.5" is misleading, as each roll results in an integer between 1 and 6. - The correct interpretation is: The average number of spaces moved is 3.5.

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Studdy Solution

STEP 1

1. The die is fair, meaning each face has an equal probability of landing face up.
2. The die has six faces numbered from 1 to 6.
3. The expected value is calculated as the average outcome over many trials.

STEP 2

1. Identify the possible outcomes and their probabilities.
2. Calculate the expected value using the formula for expected value.

STEP 3

List the possible outcomes when a die is tossed: $1, 2, 3, 4, 5, 6$ .

STEP 4

Determine the probability of each outcome. Since the die is fair, each outcome has a probability of:
$\frac{1}{6}$

STEP 5

Use the formula for expected value $E(X)$ , which is the sum of each outcome multiplied by its probability:
$E(X) = \sum_{i=1}^{6} x_i \cdot P(x_i)$
where $x_i$ are the outcomes and $P(x_i)$ is the probability of each outcome.

STEP 6

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Studdy Solution

STEP 1

1. The die is fair, meaning each face has an equal probability of landing face up.2. The die has six faces numbered from 1 to 6.3. The expected value is calculated as the average outcome over many trials.

STEP 2

1. Identify the possible outcomes and their probabilities.2. Calculate the expected value using the formula for expected value.

STEP 3

List the possible outcomes when a die is tossed: 1,2,3,4,5,6 1, 2, 3, 4, 5, 6 1,2,3,4,5,6.

STEP 4

Determine the probability of each outcome. Since the die is fair, each outcome has a probability of:16 \frac{1}{6} 61​

STEP 5

STEP 6

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The die is fair, meaning each face has an equal probability of landing face up.
2. The die has six faces numbered from 1 to 6.
3. The expected value is calculated as the average outcome over many trials.

1. Identify the possible outcomes and their probabilities.
2. Calculate the expected value using the formula for expected value.

List the possible outcomes when a die is tossed: $1, 2, 3, 4, 5, 6$ .

Determine the probability of each outcome. Since the die is fair, each outcome has a probability of:
$\frac{1}{6}$