JEE Advanced 2025 Paper 2 · Mathematics · Probability · hard

The probability that the experiment stops at an even-numbered toss is ______.

Numerical answer

A fair coin is tossed repeatedly until two consecutive heads appear for the first time. The probability that the experiment stops at an even-numbered toss is ______. Give the answer correct to two decimal places.

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Answer: 0.60

The experiment ends exactly at toss

n

if the sequence has no HH before and ends in THH (for

n\ge3

) or is HH (

n=2

). The number of such sequences is the Fibonacci number

F_{n-1}

(

1,1,2,3,5,\dots

for

n=2,3,4,\dots

; checked by brute force), so

P(N=n)=F_{n-1}/2^n

. With generating function

G(s)=\sum F_{n-1}s^n=\dfrac{s^2}{1-s-s^2}

P(\text{even})=\frac{G(\frac12)+G(-\frac12)}{2}

G(\tfrac12)=\frac{1/4}{1/4}=1

and

G(-\tfrac12)=\frac{1/4}{1+\frac12-\frac14}=\frac15

. Hence

P=\frac{1+1/5}{2}=\frac35=0.60

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