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Question

A fair coin is tossed three times. Consider the following events.
E1: Heads on the first toss.
E2: Tails on the second toss.
E3: Heads on the third toss.
E4: All three outcomes are same.
E5: Exactly one heads turns up.

Which of the following pairs/triples of events is (are) independent?

A
E1 and E4
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B
E1 and E5
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C
E1, E2 and E3
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D
E1, E2 and E4
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Solution

The correct options are
A E1 and E4
C E1, E2 and E3
E1={HHT, HTH, HTT, HHH}
P(E1)=48=12
E2={HTT, TTH, HTH, TTT}
P(E2)=48=12
E3={HTH, THH, HHH, TTH}
P(E3)=48=12
E4={HHH, TTT}
P(E4)=28=14
E5={HTT, THT, TTH}
P(E5)=38

P(E1E4)=P{HHH}=18=P(E1)P(E4), hence independent.

P(E1E5)=P{HTT}=18P(E1)P(E5), hence not independent.

P(E1E2E3)=P{HTH}=18=P(E1)P(E2)P(E3)
E1, E2, E3 are independent.

Obviously, E1E2E4=ϕ, hence not independent.

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