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:
Is ? and ? are free/independent?
Bogus, supposing that An is genuine then likelihood of second heads increments since then likelihood of having picked second coin increments and the other way around
Is ? and ? are autonomous?
Bogus, A must be valid for C to be valid, in the event that An is bogus, at that point C will consistently be bogus
Is ? and ? are autonomous given ??
Genuine, in the event that we realize reasonable coin is picked likelihood of heads is constantly 1/2 in any attempt regardless of what past outcome was.
Is ? and ? are free given ? ?
Bogus, C will consistently be reliant on An, in such a case that An is bogus then C can never be valid.
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