probility

In this exercise you will solve the following problem:

If you roll eight dice, what is the probability that each of the numbers 1

through 6 appears on top at least once?

For 1 ≦ i ≦ 6, let Ei be the event that number i doesn't show up on any of

the dice.(a) Write a formula for P(Ei).

(b) Let k ≦ 6 and 1 ≦ i1 < i2 <...< ik ≦ 6.

Write a formula for P(Ei1 ∩Ei2 ∩...∩ Eik).

(c) Now use the inclusion-exclusion formula to write a formula for

P(E1 ∪ E2 ∪ ...∪E6).

This is the probability that some number doesn't appear when you

roll eight die.

Your formula should use the summation sign, powers and binomial

coecients.

(d) Using the solution to (c), write down the probability that each of the

numbers 1 through 6 appears on top at least once (a solution in the

form of a sum is ne; it is not necessary to actually calculate the value

of the sum).