Question

Assertion :

A fair coin is tossed $3$ times. Consider the events
$A$: first toss is head; $B$: second toss is head;
$C$: exactly two consecutive heads or exactly two consecutive tails.

$A,B,C$ are independent events. Reason: $A,B,C$ are pairwise independent.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Given, a fair coin is tossed thrice
$\therefore$Total no.of outcomes=2\times 2\times2=8
A: Probability for first toss to be head,$P\left(A\right)=\frac{4}{8}=\frac{1}{2}$ (since no.of outcomes are {HTT,HHH,HHT,HTH})
B: Probability for second toss to be head,$P\left(B\right)=\frac{4}{8}=\frac{1}{2}$ (since no.of outcomes are {THT,HHH,HHT,THH})
C: Probability of two consecutive heads or tails,$P\left(C\right)=\frac{4}{8}=\frac{1}{2}$ (since no.of outcomes are {HTT,THH,HHT,TTH})
Consider,$P(A\cap B\cap C)=\frac{1}{8}$(since HHT is common events)
$P\left(A\right)\times P\left(B\right)\times P\left(C\right)=\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=\frac{1}{8}$
$P(A\cap B\cap C)=P\left(A\right)P\left(B\right)P\left(C\right)$
$\Rightarrow A,B,C$are independent events.

Consider,$P(A\cap B)=\frac{2}{8}=\frac{1}{4}$(since HHH,HHT are common events)
$P\left(A\right)\times P\left(B\right)=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$

$P(A\cap B)=P\left(A\right)P\left(B\right)$
$\Rightarrow A,B$are independent events.
Consider,$P(B\cap C)=\frac{2}{8}=\frac{1}{4}$(since THH,HHT are common events)

$P\left(C\right)\times P\left(B\right)=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$

$P(B\cap C)=P\left(B\right)P\left(C\right)$
$\Rightarrow B,C$are independent events.
Consider,$P(C\cap A)=\frac{2}{8}=\frac{1}{4}$(since HTT,HHT are common events)

$P\left(C\right)\times P\left(A\right)=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$
$P(C\cap A)=P\left(C\right)P\left(A\right)$
$\Rightarrow C,A$are independent events.
$\Rightarrow A,B,C$are pairwise independent.
But Reason is not the correct explanation for the assertion.