If the events B1, B2, . . , B, constitute a partition of the sample space S such thatP(B;) # 0 for i = 1, 2, ... , k, then for any event A of S,%3DkkP(A) = P(B;n A) =P(B;)P(A|B;).%3Di=1i=1B3B1B5B4B2...Figure 14: Partitioning the sample space S.

Answered: Image /qna-images/answer/8f7044d0-d947-453f-a2b0-3f629d7d858c.jpg

Answered: If the events B1, B2, ..., Br…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Events A₁ and A2 are mutually exclusive and form a complete partition of a sample space S with P(A₂) = 0.62. If E is an event in Swith P (E | A₁) =0.26 and P(E|A₂) = 0.18, compute P (A₂ | E) =? (Hint: Because A₁ and A2 are mutually exclusive and form a complete partition of the sample space, P (A₁) = 1 − P (A₂) ). Note: If your final answer has up to four decimal places, please enter your full answer without rounding it. If your answer contains more than four decimal places, please round it to four decimal places before entering it in the box below.
Events A₁ and A2 are mutually exclusive and form a complete partition of a sample space S with P(A₁) =0.48. If E is an event in S with P (E| A₁) =0.25 and P(E|A₂) = 0.09, compute P(A₁ | E) =? (Hint: Because A₁ and A2 are mutually exclusive and form a complete partition of the sample space, P(A₂) = 1-P (A₁)). Note: If your final answer has up to four decimal places, please enter your full answer without rounding it. If your answer contains more than four decimal places, please round it to four decimal places before entering it in the box below.
Let A and B be events in a sample space S such that S = A ∪ B. Suppose that P(A) = 0.4, P(B) = 0.8, and P(A ∩ B) = 0.2. Find each of the following. (a) P(A ∪ B) (b) P(Ac ∪ Bc )
(i) Show that if P(A) = 0 or P(A) = 1, then A is independent of every other event. Show that if A is independent of itself then P(A) is either O or 1. (ii) Suppose k events form a partition of the sample space N, i.e., they are disjoint and UL, A; = N. Assume that P(B) > 0. Prove that if P(A1|B) P(A;) for some i = 2, ..., k.
Example 2.3.23 For any three events A, B, and C defined on the sample space S, such that BCC and P(A) > 0, prove that P(B|A) s P(C|A).
For any two events A, B, and sample space S, state True or false: • If P(A) + P(B) = P(AU B) then P(AN B) = 0. • If P(AN B) = 0 then P(AU B) = P(A) + P(B). • The sample space, S, is always independent of any event A.
Events A₁ and A2 are mutually exclusive and form a complete partition of a sample space S with P (A₂) = 0.39. If E is an event in S with P (E | A₁) =0.12 and P(E|A₂) = 0.18, compute P(A₂2 | E) =? (Hint: Because A₁ and A2 are mutually exclusive and form a complete partition of the sample space, P(A₁) = 1 − P (A₂) ). = Note: If your final answer has up to four decimal places, please enter your full answer without rounding it. If your answer contains more than four decimal places, please round it to four decimal places before entering it in the box below.
Let S = {E1, E2, E3, EA} be the sample space of an experiment and let = {E1, E2}. B = {E2, E3}, and C = of the sample points are assigned as follows: P(E) P(E3) = 0.30, and P(E4) = 0.40. Find P(AU B). {Es, E,} be events from S. The probabilities 0.10. P(E2) = 0.20, A = %3D Select one: O a. 0.60 b. 0.20 O c. 0.90 O d. 0.30 e. 0.70
Let (S, P) be the sample space with S = (1, 2, 3, ..., 10) and P (a) = 1/10 for all a ES. For this sample space, define the random variables X and Y by #2. X(s) = 25 and Y(s) = s a. Evaluate P (X< 10). b. Evaluate P (Y < 10).
3. An integer is randomly chosen from the set 1, 2,..., 100. If this integer is divisible by 2 we let Y = 0, and if it is not divisible by 2 but is divisible by 3 then Y = 1. We let Y = 2 in all other cases. Find mean and variance of Y.
1a
Let A and Bevents in a sample space S such that S = AUB. Suppose that P(A) = 0.3, P(B) = 0.6,and P(AN B) = 0.2. Find P(AU B).

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If the events B1, B2, ..., Br constitute a partition of the sample space S such that P(B;) #0 for i 1, 2,..., k, then for any event A of S, k k P(A) = P(B; n A) = P(Bi)P(A|B;). i=1 i=1