Consider two independent events, A and B, where 0くP(A) < 1,0くP(8)く1. Prove that A' and B'...
1. Consider two independent events, A and B, where 0< P(A) <1,0< P(B)< 1. Prove that A and B' are independent as well.
B] <P[AP[B], and of two events A and B Give an example of two events such that P[ A with P[An B] > P[A]P[B].
Suppose two events A and B are two independent events with P(A) > P(B) and P(A U B) = 0.626 and PA กั B) 0.144, determine the values of P(A) and P(B).
Suppose that A1,A2,.., Ak are mutually exclusive events and P(B)>0. Prove that
1. Use the formula P(A) PABP(B) + P(AlBc)P(B") to prove that if P(AB) P (AlBc) then A and B are independent. Then prove the converse (that if A and B are independent then P(AIB)- P(ABe). [Assume that P(B) > 0 and P(B) > 0.]
Problem 8. Suppose that XGeom(p) and Y ~ Geom(r) are independent. Find the probability P(X <Y).
7. Define a Markov Chain on S-0,1,2,3,... with transition probabilities Pi,i+1 with 0<p < 1/2. Prove that the Markov Chain is reversible.
0.2 Question 7 (1 point) <Venn 3> There are 2 events: A, B with P(A)-0.5, P(B)-0.4, P(AUB)-0.7 Find P(BA) (2 decimal places without rounding-up) Question 8 (1 point) Saved <Venn 4>
Consider the sample space ș = {x | 0 < x < 12), and the events A = {지 2 Find the events: (a) A UB (b) AnB (c) A' n B', and (d) A' UB' x < 5} and B = {지 3 < x < 7}
4. Suppose A, B, C are events such that P(A), P(B), P(C) a. If (A, B, C) are independent, show that P(AU BUC)- b. If A, B, C are only pairwise independent, show that 17 24 SHA UBUC)<19 24