Suppose we have 10 coins such that if the i th coin is...

Suppose we have 10 coins such that if the $i$ th coin is flipped, heads will appear with probability $i / 10, i=$ $1,2, \ldots, 10 .$ When one of the coins is randomly selected and flipped, it shows heads. What is the conditional probability that it was the fifth coin?

Key Concepts

Conditional Probability

Conditional probability measures the likelihood of an event occurring given that another event has occurred. It is used to update our determination of probability in light of new knowledge, focusing on the subset of outcomes that have already been realized. This concept is essential for situations where evidence or information alters the overall probability landscape.

Bayes' Theorem

Bayes' Theorem provides a systematic way to update the probability of a hypothesis based on new evidence. It relates the conditional and marginal probabilities of random events, allowing us to reverse conditional probabilities when new data is presented. This theorem is a cornerstone of statistical inference and decision-making under uncertainty.

Law of Total Probability

The Law of Total Probability decomposes the probability of an event into several disjoint scenarios or cases. It states that the overall probability of an event can be computed by considering all different ways that event can occur, weighted by the probability of each corresponding pathway. This law is critical when considering multiple sources or causes that contribute to the overall outcome.

Biased Coins

Biased coins refer to coins that do not have an equal chance of landing on heads or tails when flipped. In probability theory, such coins have been assigned specific probabilities for heads (or tails) that differ from the classical 50/50 chance. Analyzing sequences involving biased coins is pivotal for understanding scenarios where outcomes are skewed by fixed probability weights.

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