There are 3 coins in a box. One is a two-headed coin;...

Name: Problem 39, Chapter 3: A First Course in Probability
Uploaded: 2022-02-23T05:59:59-08:00
Duration: 3 min 37 s
Channel: Philomena Marfo
Description: There are 3 coins in a box. One is a two-headed coin; another is a fair coin; and the third is a biased coin that comes up heads 75 percent of the time. When one of the 3 coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?

There are 3 coins in a box. One is a two-headed coin; another is a fair coin; and the third is a biased coin that comes up heads 75 percent of the time. When one of the 3 coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?

Key Concepts

Likelihood

Likelihood is the probability of the observed evidence under a specific hypothesis. In the context of Bayesian inference, likelihoods are used to determine how well different hypotheses explain the observed data, and they help in weighting these hypotheses accordingly.

Bayes' Theorem

Bayes' Theorem is a fundamental rule in probability theory that updates the probability for a hypothesis as more evidence or data becomes available. It combines prior belief and the likelihood of observed data to provide a revised probability, allowing decisions to be based on updated information.

Conditional Probability

Conditional probability is the measure of the probability of an event occurring given that another event has occurred. This concept is essential when dealing with events that have dependent outcomes or when additional information is revealed that affects the likelihood of different outcomes.

Prior Probability

Prior probability represents the original degree of belief in a hypothesis before new data is considered. It is crucial in Bayesian analysis, as it sets the initial probabilities that are later updated in light of the evidence.

Law of Total Probability

The Law of Total Probability states that the probability of an event can be computed as the sum of the probabilities of that event under various mutually exclusive scenarios. This ensures that all possible ways the event can occur are taken into account, which is critical for normalizing probabilities in problems involving multiple conditions.

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