Page 35 - Schaum's Outlines - Probability, Random Variables And Random Processes

PROBABILITY [CHAP 1

Check:
By counting technique, we have

1.46. There are two identical decks of cards, each possessing a distinct symbol so that the cards from
each deck can be identified. One deck of cards is laid out in a fixed order, and the other deck is
shufkd and the cards laid out one by one on top of the fixed deck. Whenever two cards with the
same symbol occur in the same position, we say that a match has occurred. Let the number of
cards in the deck be 10. Find the probability of getting a match at the first four positions.
Let A,, i = 1,2,3,4, be the events that a match occurs at the ith position. The required probability is

P(A, n A, n A, n A,)
By Eq. (1.81),

There are 10 cards that can go into position 1, only one of which matches. Thus, P(Al) = &. P(A, (A,) is
the conditional probability of a match at position 2 given a match at position 1. Now there are 9 cards left
to go into position 2, only one of which matches. Thus, P(A2 I A,) = *. In a similar fashion, we obtain
P(A3 I A, n A,) = 4 and P(A, I A, n A, n A,) = 4. Thus,

TOTAL PROBABILITY
1.47. Verify Eq. (1.44).
Since B n S = B [and using Eq. (1.43)], we have
B= B n S=Bn(A, u A, u u An)
=(B n A,) u (B n A,) u ... u (B n An)
Now the events B n A,, i = 1,2, . . . , n, are mutually exclusive, as seen from the Venn diagram of Fig. 1-14.
Then by axiom 3 of probability and Eq. (1.41), we obtain

B nA, BnA, BnA,
Fig. 1-14