(a) A gambler has a fair coin and a two-headed coin in his...

(a) A gambler has a fair coin and a two-headed coin in his pocket. He selects one of the coins at random; when he flips it, it shows heads. What is the probability that it is the fair coin? (b) Suppose that he flips the same coin a second time and, again, it shows heads. Now what is the probability that it is the fair coin? (c) Suppose that he flips the same coin a third time and it shows tails. Now what is the probability that it is the fair coin?

(a) A gambler has a fair coin and a two-headed coin in his pocket. He selects one of the coins at random; when he flips it, it shows heads. What is the probability that it is the fair coin?
(b) Suppose that he flips the same coin a second time and, again, it shows heads. Now what is the probability that it is the fair coin?
(c) Suppose that he flips the same coin a third time and it shows tails. Now what is the probability that it is the fair coin?

Key Concepts

Independent Events

Independence of events is the idea that the outcome of one event does not affect the outcome of another, assuming the conditions are fixed. In the context of coin flipping, each flip is considered independent of the others given a particular coin, which simplifies the computation of the overall likelihood of a sequence of outcomes.

Total Probability Theorem

The Total Probability Theorem helps compute the overall probability of an observed event by considering all possible ways the event can occur. This involves summing the probabilities of the event occurring under different scenarios (such as using a fair coin versus a two-headed coin), each weighted by its own probability, thereby allowing a complete assessment of the outcomes.

Conditional Probability

Conditional probability measures the probability of an event occurring given that another event has already occurred. In this problem, the probability of the coin being fair is updated based on observed outcomes from multiple flips, making conditional probability a key component of the analysis.

Bayes Theorem

Bayes Theorem is a fundamental concept in probability that relates the conditional probability of an event based on prior knowledge to likelihoods of observed data. It is particularly useful for updating the probability estimates for a hypothesis as new evidence is acquired, as seen when determining the probability that a selected coin is fair given successive coin toss outcomes.

Transcript

00:01 In part a, we want to compute the conditional probability that a coin is fair, given that a flip is a hat.

00:13 So this equals the probability that a coin is fair and a flip is a head, divided by the probability that a flip is head.

00:33 So this equals the probability that a conditional probability that a flip is head.

00:41 That a flip is head given that the coin is fear times the probability that the coin is fair.

00:51 And if a flip is head, there are two cases.

00:55 The first case is that a coin is fear.

01:06 And the second case is that the coin is two headed.

01:25 So if a coin is fear, the probability that a flip is head is a hub.

01:32 And the probability that a coin fear is also a hub.

01:42 If a coin is two -headed, a probability that a flip is head is one, and the probability that the coin is two -headed is a hot.

01:55 So this equals one -third.

01:59 In part b, we want to compute the conditional probability that a coin in the fear given that both flips are half.

02:12 So we use the head squared and still equals the probability that two flips are heads given that the coin sphere times the probability that a coin is a fear divided by the probability that two flips a heads given that the quality is a fear times the probability that the coin is fear.

02:51 And for the second case, it's the same...

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