At time 0, a coin that comes up heads with probability p is...

At time $0,$ a coin that comes up heads with probability $p$ is flipped and falls to the ground. Suppose it lands on heads. At times chosen according to a Poisson process with rate $\lambda,$ the coin is picked up and flipped. (Between these times the coin remains on the ground.) What is the probability that the coin is on its head side at time $t ?$ Hint What would be the conditional probability if there were no additional flips by time $t,$ and what would it be if there were additional flips by time $t ?$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Conditional Probability

Conditional probability is used to analyze the outcome based on whether or not additional flips occur. In this context, the coin's state at time t depends on two mutually exclusive scenarios: one in which no extra flips occur (and the coin remains in its initial state) and one in which at least one flip occurs and the state is determined by the outcome of the most recent independent flip.

Poisson Process

A Poisson process describes the random occurrence of events over time, where the number of events in a fixed interval follows a Poisson distribution. The coin flips occur at times dictated by a Poisson process with rate ?, meaning the probability of no additional flip by time t is given by exp(–?t), a crucial element for computing the overall probability of the coin's state.

Bernoulli Trial

Each coin flip is modeled as a Bernoulli trial – an experiment with exactly two possible outcomes (heads or tails) that occur with fixed probabilities (p for heads and 1–p for tails). This concept is fundamental for determining the probability that the coin lands on heads after any additional flip, independent of previous flips.

Transcript

Please give Ace some feedback