For each of the following pairs of events, A and B, determine whether A and B are dependent or not. Show your calculations and briefly explain. (a) We have a deck of 52 playing cards, from which we draw two cards in turn. Let A denote the event that the first card we draw is a King. Let B denote the event that the second card we draw is a Queen. (b) We have one fair die (also called a dice). Each of the six faces come up with equal probabilities 1/6. We throw the dice two times in a row. Let A denote the event that the first throw yields a six. Let B denote the event that the second throw yields a six. (c) We draw a bit-string x of length 4 uniformly at random among all bit-strings of length 4. If the hamming weight of x is even, then we draw a bit-string y of length 6 uniformly at random among all bit-strings of length 6. If the hamming weight of x is odd, then we draw a bit-string y of length 42 uniformly at random among all bit-strings of length 42. Let A denote the event that the hamming weight of x is even. Let B denote the event that the hamming weight of y is even.
For each of the following pairs of events, A and B, determine whether A and B are dependent or not.
Show your calculations and briefly explain.
(a) We have a deck of 52 playing cards, from which we draw two cards in turn. Let A denote the event
that the first card we draw is a King. Let B denote the event that the second card we draw is a
Queen.
(b) We have one fair die (also called a dice). Each of the six faces come up with equal probabilities 1/6.
We throw the dice two times in a row. Let A denote the event that the first throw yields a six. Let
B denote the event that the second throw yields a six.
(c) We draw a bit-string x of length 4 uniformly at random among all bit-strings of length 4. If the
hamming weight of x is even, then we draw a bit-string y of length 6 uniformly at random among all
bit-strings of length 6. If the hamming weight of x is odd, then we draw a bit-string y of length 42
uniformly at random among all bit-strings of length 42. Let A denote the event that the hamming
weight of x is even. Let B denote the event that the hamming weight of y is even.
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