Page 43 - Probability, Random Variables and Random Processes

CHAP. 11 PROBABILITY

Let A, be the event that point of k occurs before 7. Then
P(C) = P(A,)P(point = k)
ke(4, 5, 6, 8, 9. 10)
By Eq. (1.89) (Prob. 1.62),

Again from Fig. 1-3,

Now by Eq. (1 .go),

Using these values, we obtain

Supplementary Problems

1.64. Consider the experiment of selecting items from a group consisting of three items (a, b, c).
(a) Find the sample space S, of the experiment in which two items are selected without replacement.
(b) Find the sample space S, of the experiment in which two items are selected with replacement.
Ans. (a) S, = {ab, ac, ba, bc, ca, ch)
(b) S, = {aa, ah, ac, ha, bh, bc, ca, cb, cc}
1.65. Let A and B be arbitrary events. Then show that A c B if and on1.y if A u B = B.
Hint : Draw a Venn diagram.

1.66. Let A and B be events in the sample space S. Show that if A c B, then B c A.
Hint: Draw a Venn diagram.

1.67. Verify Eq. (1.1 3).
Hint: Draw a Venn diagram.

1.68. Let A and B be any two events in S. The difference of B and A, denoted by B - A, is defined as
B-A=BnA
The symmetric difference of A and B, denoted by A A B, is defined by
A A B = (A - B) U (B -- A)
Show that
--
A A B =(A u B) n (A 1-1 B)
Hint: Draw a Venn diagram.