Page 49 - DMTH404_STATISTICS

Unit 3: Conditional Probability and Independence Baye’s Theorem

c
(i) We want P(A  A  A ). By Remark 4, we can write Notes
1 2 3
c
P(A  A  A ) P(A )P(A )P(A )
c

1 2 3 1 2 3
9 1 1
 . .  0.009.
10 10 10
(ii) We want to find,the probability of
c
(A  A  A )  (A  A  A ) (A A A ).
c
c
1
3
3
2
2
2
1
1
3
Notice that these events are disjoint and that each has the probability 0.009 (see (i)). Hence,
the required probability is
Example 25: The probability that a person A will be alive 20 years hence is 0.7 and the
probability that another person B will be alive 20 years hence is 0.5. Assuming independence,
let’s find the probability that neither of them will be alive after 20 years.
The probability that A dies before twenty years have elapsed is 0.3 and the corresponding
probability for B is 0.5. Hence the probability that neither of them will be alive 20 years hence
is
0.3 × 0.5 = 0.15,
by virtue of independence.
We now give you some exercises based on the concept of independence.

3.4 Repeated Experiments and Trials

We must mention that we have earlier discussed rolls of two dice or three or more tosses of a
coin without bringingin the concept of repeated trials. The following discussion is only an
elementary introduction to the topic of repeated trials.
To fix ideas, consider the simple experiment of tossing a coin twice. The sample space
corresponding to the first toss is S = {H, T} say, where H = Head, T = Tail. Similarly the sample
1
space S for the second toss is also {H, T}. Now observe that the sample space for two tosses is
2
 = {(H, H), (H, T), (T, H), (T, T)}, where (H, H) stands for head on first toss followed by a head
on the second toss. The pairs (H, T), etc. are also similarly defined. Note that  consists of all
ordered pairs that can be formed by choosing a point from S followed by a point from S .
1 2
Mathematically we say that  is the Cartesian product S × S (read, S cross S ) of S and S .
1 2 1 2 1 2
Now consider an experiment of tossing a coin and then rolling a die. The sample space
corresponding to toss of the coin is S = (H, T) and that corresponding to the roll of the die is
1
S = (1, 2, 3, 4, 5, 6). The sample space of the combined experiment is
2
W = {(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6),
(T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6) = S × S
1 2.
Taking a cue from these two examples we can say that if S and S are the sample spaces for two
1 2
random experiments  and  then the Cartesian product S × S is the sample space of the
1 2 1 2
experiment consistihg of both  and  .
1 2
Sometimes we refer to S × S as the product space of the two experiments.
1 2

LOVELY PROFESSIONAL UNIVERSITY 41