# Probability

Suppose that we have a box that contains two coins:

A fair coin: P(H)=P(T)=0.5 .

A two-headed coin: P(H)=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 A : first coin toss is H .

Event B : second coin toss is H .

Event C : two coin tosses result in HH .

Event D : the fair coin is chosen.

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

A and B are independent.

True
False

A and C are independent.

True
False

A and B are independent given D .

True
False

A and C are independent given D .

True
False

Suppose three random variables X , Y , Z have a joint distribution

PX,Y,Z(x,y,z)=PX(x)PZ∣X(z∣x)PY∣Z(y∣z).
Then, X and Y are independent given Z .

True
False

Suppose random variables X and Y are independent given Z , then the joint distribution must be of the form

PX,Y,Z(x,y,z)=h(x,z)g(y,z),
where h,g are some functions.

True
False

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