If a coin is tossed a sequence of times, what is the probability that the first head will occur after the fifth toss, given that it has not occurred in the first two tosses?

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Solution: C

Let [math]T[/math] equal the number of tosses until the first head appears. [math]T[/math] has distribution [math]P(T=k) = (1/2)^k [/math] for [math]k\geq 1 [/math]. The probability that the first head appears on the fifth toss, given that it has not appeared in the first two tosses, equals

[[math]] P(T=5|T\gt2) = \frac{P(T=5)}{P(T\gt2)} = \frac{(1/2)^5}{1-0.5-0.25} = \frac{(1/2)^5}{(1/2)^2} = (1/2)^3 = 0.125. [[/math]]