Your a-priori probability that the coin is biased is 0 . Prove that no matter how many times you

Your a-priori probability that the coin is biased is 0 ....

Key Concepts

Bayesian Inference

Bayesian inference is the process of updating beliefs or probability estimates for a hypothesis based on new data, using Bayes’ theorem. It integrates prior beliefs with the likelihood of observing the new data to produce a posterior probability, which reflects both the prior and the new evidence.

Prior Probability

Prior probability is the initial degree of belief in a hypothesis before new evidence is considered. In the context of Bayesian analysis, if a hypothesis is assigned a prior probability of zero, it mathematically excludes the hypothesis from being supported by any amount of new evidence since multiplication by zero yields zero.

Posterior Probability

Posterior probability is the updated probability of a hypothesis after considering the new evidence via Bayes’ theorem. When the prior probability is zero, the resulting posterior will also be zero, demonstrating that no amount of new data or evidence can compensate for an initial belief of impossibility.

Zero Prior Problem

The zero prior problem highlights the issue in Bayesian updating when a hypothesis is given a prior probability of zero. It illustrates that the mathematical framework of Bayesian inference does not allow the probability of such a hypothesis to increase, no matter how strong the supporting evidence is, underscoring the critical role of careful prior selection in Bayesian analysis.

Transcript

00:01 In this problem, it is said that a coin is biased in such a way that a head is twice as likely to occur as a tail.

00:08 We need to find the expected number of tails when this coin is twice.

00:12 Now, first of all, if this coin has twice the probability of obtaining a head as the probability of obtaining a tail, then that will mean that the probability of obtaining a head will be two times the probability of obtaining a tail.

00:28 And since the probability of obtaining a head and the probability of obtaining a tails should the sum of these should be equal to one because when you toss a coin you will either get a head or a tail so the sum of these two probabilities should be one probability of heads is 2 pt so 2 pt plus pt is equal to 1 that implies that 3 pt is equal to 1 which means that the probability of obtaining a tails is is 1 divided by 3 and thus the probability of getting our heads will be 2 pt which is 2 times 1 by 3 which is 2 divided by 3 now let us consider x to be the random variable of obtaining a tail so the probability of x equals to 0 means the probability of getting 0 p's so that means we will have a head on the first throw and a head on the second toss of the coin so that will be 2 by 3 times 2 by 3 because the probability of obtaining ahead on the first toss is 2 by 3.

01:36 The probability on obtaining ahead on the second toss is 2 by 3.

01:39 So if we take both of these events together, we use the multiplication rule to multiply them and we will get 4 by 9.

01:47 Next, the probability of x equals to 1 is the probability of getting 1 tail.

01:52 So one possibility is getting ahead on the first coin and the tail of the tail of the tail.

01:59 On the second toss.

02:00 So the probability will be 2 by 3 times 1 by 3.

02:03 And there is another option which is a tail on the first toss and a head on the second toss.

02:09 So we add the probabilities of these two options in order to get the total probability...

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