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Waiting for first tails

ProbabilityEasy

A biased coin lands heads with probability $\frac{3}{4}$ on each independent toss. It is tossed until the first tails appears. What is the expected number of tosses, counting the toss that shows tails?

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Hint

The number of tosses is geometric; use the probability of tails as the success probability.

Solution

Each toss shows tails with probability $\frac{1}{4}$ , independently of the others.
The number of tosses until the first tails is geometric with success probability $\frac{1}{4}$ .
The expected value of a geometric variable is $1$ divided by the success probability, here $1 \div (\frac{1}{4})$ .
The expected number of tosses is $4$ .

Answer: $4$

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