Chapter 7: Q 7.6 (page 352)

A fair die is rolled $10$ times. Calculate the expected sum of the $10$ rolls.

Short Answer

Expert verified

The expected sum of the $10$ rolls value are $35$ .

Step by step solution

Given Information

Calculate the expected sum of the $10$ rolls.

Let random variable $X_{i}$ represent the number on the face of the dice, after the roll.

Explanation

Since there are $6$ numbers on a dice, the probability of getting any of the numbers is

$P (X_{i} = 1) = \frac{1}{6},$

$P (X_{i} = 2) = \frac{1}{6}$ ,

$P (X_{i} = 3) = \frac{1}{6}$ ,

$P (X_{i} = 4) = \frac{1}{6}$ ,

$P (X_{i} = 5) = \frac{1}{6}$ .

$P (X_{i} = 6) = \frac{1}{6}$ .

Expected sum of the 10 rolls

Let us find the expected sum of the $10$ rolls.

$E (X_{i}) = 1 \times P (X_{i} = 1) + 2 \times P (X_{i} = 2)$

role="math" localid="1647251498640" $+ 3 \times P (X_{i} = 3) + 4 \times P (X_{i} = 4)$

role="math" localid="1647251538454" $+ 5 \times P (X_{i} = 5) + 6 \times P (X_{i} = 6)$

Substitute the value,

$= 1 \times \frac{1}{6} + 2 \times \frac{1}{6}$

$+ 3 \times \frac{1}{6} + 4 \times \frac{1}{6}$

$+ 5 \times \frac{1}{6} + 6 \times \frac{1}{6}$ .

Multiply the value

Multiply the value,

$= \frac{1}{6} \times (1 + 2 + 3 + 4 + 5 + 6)$

$= \frac{21}{6}$

Divide

$= \frac{7}{2}$ .

Substitute the value

Therefore, the expected sum of the $10$ rolls is,

$E (X) = 10 \times E (X_{i})$

Substitute,

$= 10 \times \frac{7}{2}$

$= 35$ .

Final answer

The expected sum of the $10$ rolls value are $35$ .

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A fair die is rolled $10$ times. Calculate the expected sum of the $10$ rolls.

Short Answer

Step by step solution

Given Information

Explanation

Expected sum of the 10 rolls

Multiply the value

Substitute the value

Final answer

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A fair die is rolled 10times. Calculate the expected sum of the10 rolls.

Short Answer

Step by step solution

Given Information

Explanation

Expected sum of the 10 rolls

Multiply the value

Substitute the value

Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

Geometry

Discrete Mathematics

Calculus

Pure Maths

Study anywhere. Anytime. Across all devices.

A fair die is rolled $10$ times. Calculate the expected sum of the $10$ rolls.