Suppose that A1,A2,.., Ak are mutually exclusive events and P(B)>0. Prove that
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).
(a) Prove that if A1, A2, . . . , are mutually exclusive, then P(An) → 0 as n → oo. (Recall that whenever Σοοι pn is finite and all the pn's are nonnegative, then Pn-+0 as n o.) (b) Suppose 1 flip a fair coin forever. Let An be the event that the rnth flip is a head. Since the coin is fair, P(An)-.Notice that P(An) 0 asnoo. How, then, can the previous problem still be true? Each An...
i. (2nd Principle of Induction): Suppose that a1 = 2 and a2 = 4 and for n > 2, an = 5an-1 – 6an-2. Prove that for all n e N, an = 2". (This is easy. Show precisely where you need the 2nd Principle.)
The prior probabilities for events A1 and A2 are P(A1) = .50 and P(A2) = .50. It is also known that P(A1 A2) = 0. Suppose P(B | A1) = .10 and P(B | A2) = .04. Are events A1 and A2 mutually exclusive? Compute P(A1 B) (to 4 decimals). Compute P(A2 B) (to 4 decimals). Compute P(B) (to 4 decimals). Apply Bayes' theorem to compute P(A1 | B) (to 4 decimals). Also apply Bayes' theorem to compute P(A2 |...
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.]
Let s = {k=1CkXAz be a simple function, where {A1, A2, ... , An} are disjoint. Prove that for every p>0, |CK|PXAR
The prior probabilities for events A1 and A2 are P(A1) = 0.45 and P(A2) = 0.50. It is also known that P(A1 ∩ A2) = 0. Suppose P(B | A1) = 0.20 and P(B | A2) = 0.05. If needed, round your answers to three decimal digits. a) Are A1 and A2 mutually exclusive? b) Compute P(A1 ∩ B) and P(A2 ∩ B). c) Compute P(B). d) Apply Bayes’ theorem to compute P(A1 | B) and P(A2 | B).
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].
The prior probabilities for events A1 and A2 are P(A1) = 0.30 and P(A2) = 0.40. It is also known that P(A1 ∩ A2) = 0. Suppose P(B | A1) = 0.20 and P(B | A2) = 0.05. If needed, round your answers to three decimal digits. (a) Are A1 and A2 mutually exclusive? Explain your answer. The input in the box below will not be graded, but may be reviewed and considered by your instructor. (b) Compute P(A1 ∩...
The prior probabilities for events A1 and A2 are P(A1) = 0.40 and P(A2) = 0.45. It is also known that P(A1 ∩ A2) = 0. Suppose P(B | A1) = 0.20 and P(B | A2) = 0.05. If needed, round your answers to three decimal digits. (a) Are A1 and A2 mutually exclusive? - Select your answer -YesNoItem 1 Explain your answer. The input in the box below will not be graded, but may be reviewed and considered by...