778 CHAPTER 38. PROBABILITY
8. Let n be a natural number and let k1 + k2 + · · ·kr = n where ki is a non negativeinteger. The symbol (
nk1k2 · · ·kr
)denotes the number of ways of selecting r subsets of {1, · · · ,n}which contain k1,k2 · · ·krelements in them. Find a formula for this number.
9. Is it ever the case that (a+b)n = an +bn for a and b positive real numbers?
10. Is it ever the case that√
a2 +b2 = a+b for a and b positive real numbers?
11. Is it ever the case that 1x+y =
1x +
1y for x and y positive real numbers?
12. Derive a formula for the multinomial expansion,(∑
pk=1 ak
)n which is analogous tothe binomial expansion. Hint: See Problem 8.
13. Let X be a binomial random variable. Thus P(X = k) =
(nk
)pkqn−k where p is
the probability of success and q = 1− p is the probability of failure. The expectedvalue of X denoted as E (X) , is defined as
n
∑k=0
kP(X = k) .
Show the expected value of X equals np.
14. The variance of the random variable in the above problem is defined as
σ2 ≡
n
∑k=0
(k−E (X))2 P(X = k)
Find σ2. You should get npq.
15. Find the probability of drawing from a shuffled deck of playing cards four hearts.Hint: Use principles of counting to find the number of ways of drawing four hearts.There are 13 of these. Now how many ways can you pull out four of them? Thennote there are 52 cards in all. How many ways can you pull out four cards from these.
16. Find the probability of obtaining 2 clubs and three spades from a shuffled deck ofcards.
17. A pond has N fish and 120 of these are marked fish. What is the probability in termsof N of catching 10 fish, two of which are marked and 8 of which are unmarked?
18. Show that in general, for k ≤ m < N
k
∑j=0
(mj
)(N−mk− j
)(
Nk
) = 1