Page 16 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 11 PROBABILITY

We may also extend the definition of independence to more than three events. The events A,, A,, . . . ,
A, are independent if and only if for every subset (A,,, A,, , . . . , A,,) (2 5 k 5 n) of these events,
P(Ail n A,, n . . n Aik) = P(Ai1)P(Ai,) P(Aik) (1.51)
Finally, we define an infinite set of events to be independent if and only if every finite subset of these
events is independent.
To distinguish between the mutual exclusiveness (or disjointness) and independence of a collec-
tion of events we summarize as follows:
1. If (A,, i = 1,2, . . . , n} is a sequence of mutually exclusive events, then

P( i) A,) = P(AJ
i= 1 i= 1
2. If {A,, i = 1,2, . . . , n) is a sequence of independent events, then

and a similar equality holds for any subcollection of the events.

Solved Problems

SAMPLE SPACE AND EVENTS
1.1. Consider a random experiment of tossing a coin three times.
(a) Find the sample space S, if we wish to observe the exact sequences of heads and tails
obtained.
(b) Find the sample space S, if we wish to observe the number of heads in the three tosses.

(a) The sampling space S, is given by
S, = (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT)
where, for example, HTH indicates a head on the first and third throws and a tail on the second
throw. There are eight sample points in S,.
(b) The sampling space S, is given by
Sz = (0, 1, 2, 3)
where, for example, the outcome 2 indicates that two heads were obtained in the three tosses. The
sample space S, contains four sample points.

1.2. Consider an experiment of drawing two cards at random from a bag containing four cards
marked with the integers 1 through 4.
(a) Find the sample space S, of the experiment if the first card is replaced before the second is
drawn.
(b) Find the sample space S, of the experiment if the first card is not replaced.
i
(a) The sample space S, contains 16 ordered pairs (i, J], 1 I 1 5 j 5 4, where the first number
4,
1
indicates the first number drawn. Thus,
[(l, 1) (1, 2) (1, 3) (1,4))