Question

Suppose that we have a box that contains two coins:

A fair coin: ?(?)=?(?)=0.5 .

A two-headed coin: ?(?)=1 .

A coin is chosen at random from the box, i.e. either coin is chosen with probability 1/2 , and tossed twice.
Conditioned on the identity of the coin, the two tosses are independent.

Define the following events:

Event ? : first coin toss is ? .

Event ? : second coin toss is ? .

Event ? : two coin tosses result in ?? .

Event ? : the fair coin is chosen.

For the following statements, decide whether they are true or false.

? and ? are independent?

True or False, also explanation???

? and ? are independent?
True or False, also explanation.

? and ? are independent given ??
True or False, also explanation.

? and ? are independent given ? ?
True or False, also explanation.

Answer 1

? and ? are independent?

false, because if A is true then probability of second heads increases because then proability of haviing chosen second coin increases and vice versa

? and ? are independent?

false, A has to be true for C to be true, if A is false then C will always be false

? and ? are independent given ??

true, if we know fair coin is chosen probility of heads is always 1/2 in any try no matter what previous result was.

? and ? are independent given ? ?

false, C will always be dependent on A, because if A is false then C can never be true.

P.S. (please upvote if you find the answer satisfactory)