Question

We toss a fair coin twice, define A ={the first toss is head}, B ={the second toss is tail}, and C={the two tosses have different outcomes}. Show that events A, B, and C are pairwise independent but not mutually independent.

Answer #1

We are given here that:

A ={the first toss is head},

B ={the second toss is tail},

C={the two tosses have different outcomes}

P( A and B) = P( first coin id heads and second toss is tail) = 0.5*0.5 = 0.25

P(A) = P(B) = 0.5

Therefore, P(A and B) = P(A)P(B) = 0.25

**Therefore A, B are pairwise independent
events.**

P(C) = P(HT) + P(TH) = 0.25 + 0.25 = 0.5

P(A and C) = P(first coin toss is head and second is tail) =
0.25 = P(A)P(C)

**Therefore A and C are pairwise independent.**

P(B and C) = P(second coin is tail and first is heads) = 0.25 =
P(B)P(C)

**Therefore B and C are pairwise independent
events.**

P(A and B and C) = P(first toss is heads and second is tail) = 0.25 which is not equal to P(A)P(B)P(C) = 0.125

**Therefore: A, B and C are not mutually
independent.**

A coin is tossed twice. Consider the following events.
A: Heads on the first toss.
B: Heads on the second toss.
C: The two tosses come out the same.
(a) Show that A, B, C are pairwise independent but not
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(b) Show that C is independent of A and B but not of A ∩ B.

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A
A and B are independent
B
A and B are disjoint
C
The probability of their intersection is P(A)P(B)
D
P(A/B)=P(B/A)

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Conditioned on the identity of the coin, the two tosses are
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Define the following events:
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If you can show steps, that'd be great. I'm not fully sure what
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