Two fair dice are rolled. Let B = the event that the sum is odd, A = the event that the sum is 7 or more, and C= the event that at least one die shows a 2. For these three events, which pairs of events are independent and which pairs are dependent? ..... How can pairs of events be shown to be independent? O A. Events E and F are independent when P(E|F) = P(E) and P(F|E) # P(F). O B. Events E and F are independent when P(E|F) # P(E) and P(F|E)#P(F). OC. Events E and F are independent when P(E|F) = P(E) and P(F|E) = P(F). O D. Events E and F are independent when P(E|F)#P(E) and P(F|E) = P(F). Which pairs of events are independent? Select all that apply. O A. events A and C O B. events B and C O C. events B and A O D. No pairs are independent. Which pairs of events are dependent? Select all that apply. O A. events B and C O B. events A and C O C. events B and A D. No pairs are dependent. O O O O

If events A and B are independent, then P(A U B) = 0. True or false?

Given 2 events, A and B. Suppose P(A) =0.6 and P(B)=0.6. Which one of the following statements must be correct? Select one: O OO O O a Event A and Event B are the same event. b. Event A and Event B can not be mutually exclusive c. P(A and B) 0.6

Flip two fair coins. Then the sample space is {HH, HT, TH, TT} where T = tails and H = heads. The outcomes HT and TH are different and the HT means that the first coin showed heads and the second coin showed tails. The TH means that the first coin showed tails and the second coin showed heads. Let A = the event of getting at least one tail while letting B = the event of getting all tails. What is the probability of event A ∩ B?   a. 0.66   b. 0.50   c. 0.25   d. 0.75

Let A and B be two events with P(A and Bc) = 1. Find P(B). Recall that Bc denotes the complement of the event B.

(a) For each one of the statements below say whether the statement is true or false, explaining your answer. i. Let A and B be two events such that P(A) > 0 and P(B) > 0. If it holds that P(A|B) = 0, then P(B|A) = 1. 4 i. For two independent events A and B such that P(A) > 0) and P(B) > 0, then: P(AUB) = P(A) = P(B). iii. For a random variable X, then: Var(X)> Var(X + 2). iv. A random sample of size n is a sequence of n independent and non-identically distributed random variables. v. The standard normal distribution has fatter tails than the Student's t distribution with finite degrees of freedom. istribution with

Let A and B be two events such that the probability of B is 0.5 and the probability of both is 0.2. Find the probability of A given that B occurs.

Event E and event F are independent. Fiven P(E) = 0.3, P = (F) 0.5, compute P(E and F) and P(E or F). a) P(E and F) = b ) P(E or F)

Suppose that E and F are two events and that P(E and F)=0.2 and P(E)=0.4. What is P(F|E)​? P(F|E)​= ______