Discrete Mathematics for Computer Science

Exercises 517

Hence, from Bayes' Theorem and the Theorem of Total Probability,

P(A 2 I B) = P(B IAD) P(A)

P(B)

P(B I A 2 ). P(A 2 )

P(B I Al). P(A1) + P(B I A 2 ). P(A 2 ) + P(B I A 3 ) .P(A 3 )

(0.08)(0.3)

(0.1)(0.2) + (0.08)(0.3) + (0.03)(0.5)

ONU

W Exercises

1. A sample space 02 = {Wl,09........ (06} has the following probability density:

P(0o9) = 2/5 P(094) = 1/10

P(092) = 1/10 P(095) = 1/10

P((03) = 1/5 P(0)6) = 1/10

Which of the following pairs A, B of events are independent? Explain your answer.

(a) A = {w0 1 , oa 2 ) and B = {oa 3 , W 5 }

(b) A = {091, W061 and B = {f02, (061

(c) A = 0 and B = {o93, (04, 0o5}

(d) A = {09 1 , 03 } and B = {W 3 ,60 4 , a) 5 }

2. Under which of the following circumstances is the pair A, B of events in sample space
   Q an independent pair? Explain your answer.

(a) A and B are disjoint, P(A) > 0, and P(B) > 0

(b) P(A)= 0 and P(B) > 0

(c) P(A) = P(B) = 0

3. Suppose A and B are disjoint events in a sample space 02. Is it possible that A and B
   could be independent? Explain your answer.


4. A fair die is rolled, and a fair coin is tossed. The sample space is taken to be Q? =
   S21 x S22 where 0Ž1 is the six-element sample space for the die and Q?2 is the two-
   element sample space for the coin. Let A c Q 1 be the event "a^5 is rolled." Let B C_ Q2
   be the event "heads." Let C C f2 be the event "at most two spots on the top face of
   the die (with heads or tails on the coin) or at least five spots on the top face of the
   die together with heads on the coin." Let D be the event "at least a^5 on the die (with
   heads or tails on the coin)." Which of the following sets of events are independent
   sets? Explain your answer.
   (a) {A, B}
   (b) IA, B, C}
   (c) {B, C}
   (d) {B, C, D}