JEE Main 2023 · Mathematics · Probability · hard

A fair coin is tossed repeatedly until two consecutive heads appear for the first time. Given that this happens within the first 6 tosses, the…

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A fair coin is tossed repeatedly until two consecutive heads appear for the first time. Given that this happens within the first

6

tosses, the probability that it happens exactly on the

6^{\text{th}}

toss is

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Answer: (B) $\dfrac{5}{43}$

Count sequences in which $HH$ first appears at the end of toss

n

: the sequence must end in $THH$ (for

n\ge3

) with no earlier $HH$, and the counts follow the Fibonacci pattern

1,1,2,3,5

for

n=2,3,4,5,6

. Their probabilities are

\frac{c_n}{2^n}

, so

P(\text{within }6)=\frac14+\frac18+\frac2{16}+\frac3{32}+\frac5{64}=\frac{16+8+8+6+5}{64}=\frac{43}{64}

. Also

P(\text{exactly }6)=\frac5{64}

. The conditional probability is

\frac{5/64}{43/64}=\frac5{43}

. A direct enumeration of all

2^6

outcomes confirms these counts.

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