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?

• 0.075
• 0.1
• 0.125
• 0.15
• 0.175

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}{\sum_{k \geq 3} (1/2)^k } = \frac{(1/2)^5}{(1/2)^2} = (1/2)^3 = 0.125. [[/math]]