Question

A fair die is rolled 10 times. Find the expected value of:a) the sum of the numbers in the ten rolls;b) the number of multiples of three in the ten rolls;c) the number of different faces that appear in the ten rolls. (Hint:Use the indicator method.LetAibe the event that theith face appears at least once.)

Answer 1

Answer:

a) 3.5

b) 3.33

c) $6 - 6\left(\begin{array}{ccc}5\\6\end{array}\right)^{10}$

Step-by-step explanation:

As given,

A fair die is rolled 10 times

a)

Expected value of Sum of the number in 10 rolls = $\frac{1}{6}(1+2+3+4+5+6) = \frac{21}{6}$

                                                                                = 3.5

∴ we get

Expected value of Sum of the number in 10 rolls = 3.5

b)

Ley Y : number of multiples of 3

Y be Binomial

Y - B(n = 10, p = $\frac{2}{6}$ )

Now,

Expected value = E(Y) = np = 10×$\frac{2}{6}$  = 3.33

c)

Let m = total number of faces in a die

⇒m = 6

As die is roll 10 times

⇒n = 10

Now,

Let Y = number of different faces appears

Now,

Expected value, E(Y) = m - m$\left(\begin{array}{ccc}m-1\\m\end{array}\right)^{n}$

                                  = $6 - 6\left(\begin{array}{ccc}6-1\\6\end{array}\right)^{10} = 6 - 6\left(\begin{array}{ccc}5\\6\end{array}\right)^{10}$