A coin having probability p of coming up heads is...

A coin having probability $p$ of coming up heads is continually flipped until both heads and tails have appeared. Find
(a) the expected number of flips;
(b) the probability that the last flip lands heads.

Key Concepts

Law of Total Probability

The law of total probability allows one to compute the overall probability of an event by considering all possible ways the event can occur, each weighted by the probability of the conditioning event. This concept is essential when determining probabilities such as the chance that the last flip lands on heads by conditioning on the first flip outcome.

Stopping Time

A stopping time is the random time at which a given process is terminated based on a pre-defined condition. In this problem, the stopping time refers to the moment when both heads and tails have appeared, thereby ending the flipping sequence.

Linearity of Expectation

Linearity of expectation is a principle that allows the expected value of a sum of random variables to be calculated as the sum of their individual expected values, regardless of whether they are independent. It underpins the calculation of the total number of flips by adding the first flip to the expected additional flips needed.

Geometric Distribution

The geometric distribution models the number of independent trials required for a single success in a sequence of Bernoulli trials. In this context, after the first coin flip, one waits for the opposite outcome to occur, which follows a geometric distribution with a success parameter equal to the probability of obtaining the required outcome.

Conditional Expectation

Conditional expectation involves breaking a problem into several cases based on the outcome of an initial event and then weighting the expected values of those cases by their probabilities. This approach is used in the coin flip problem by considering the case when the first flip is heads and when it is tails.

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