Advanced High-School Mathematics

332 CHAPTER 6 Inferential Statistics

E(X) = E(X 1 +X 2 +X 3 ) = E(X 1 )+E(X 2 )+E(X 3 ) = 1+

3

2

+3 =

11

2

.

The generalization of the above problem to that of finding the ex-
pected number of boxes needed to purchase before collecting all ofn
different prizes should now be routine! The answer in this case is

E(X) = 1 +

n

n− 1

+

n

n− 2

+···+

n

2

+n=n

∑n

k=1

1

k

.

Generalization 3: Fixed sequences of binary outcomes

In section 6.1.4 we considered the experiment in which a coin is repeat-
edly tossed with the random variableXmeasuring the number of times
before the first occurrence of a head. In the present section we modify
this to ask such questions such as:

• what is the expected number of trials before obtaining two heads
  in a row?, or


• what is the expected number of trials before seeing the sequence
  HT?

What makes the above questions interesting is that on any two tosses
of a fair coin, whereas the probability of obtaining the sequencesHH
andHT are the same, the expected waiting times before seeing these
sequences differ. The methods employed here can, in principle, be
applied to the study of any pre-determined sequence of “heads” and
“tails.”