Chapter 17 Conditional Probability734
LetDbe the event thatan even number ofcoins are heads.(a)Use the four step method to determine the probability space for this experiment
and the probability of each ofA;B;C;D.
(b)Show that these events are not mutually independent.(c)Show that they are 3-way independent.Problem 17.24.
LetA;B;Cbe events. For each of the following statements, prove it or give a
counterexample.
(a)IfAis independent ofB, thenAis also independent ofB.(b)IfAis independent ofB, andAis independent ofC, thenAis independent of
B\C.
Hint:ChooseA;B;Cpairwise but not 3-way independent.
(c)IfAis independent ofB, andAis independent ofC, thenAis independent of
B[C.
Hint:Part (b).
(d)IfAis independent ofB, andAis independent ofC, andAis independent of
B\C, thenAis independent ofB[C.
Problem 17.25.
LetA;B;C;Dbe events. Describe counterexamples showing that the following
claims are false.
(a)
False Claim.IfAandBare independent givenC, and are also independent given
D, thenAandBare independent givenC[D.
(b)
False Claim.IfAandBare independent givenC, and are also independent given
D, thenAandBare independent givenC\D.
Hint:ChooseA;B;C;D3-way but not 4-way independent.
soAandBare not independent givenC\D.