Solving a Probability Problem Involving Coin Tosses

A gambler has in his pocket a fair coin and a two-headed coin. He selects one of the coins at random, and when he flips it, it shows heads. What is the probability that it is the fair coin? 

Probability is a branch of mathematics that is used to study random events. It is used to calculate the likelihood of a particular event occurring. One common example of a random event is the toss of a coin. In this blog post, we will solve a probability problem involving coin tosses. We will use conditional probability to find the probability of selecting a fair coin given that the coin shows heads.

Let's begin by defining some terms that will help us solve the problem.

Let F be the event that the fair coin is selected, and T be the event that the two-headed coin is selected.

Let H be the event that the coin shows heads, and let T be the event that the coin shows tails.

(a) To find the probability that the gambler selects the fair coin given that the coin shows heads, we need to use Bayes' theorem. Bayes' theorem states that the probability of an event given some evidence is proportional to the likelihood of the evidence given the event. Using Bayes' theorem, we can write:

P(F|H) = P(H|F) * P(F) / (P(H|F) * P(F) + P(H|T) * P(T))

where P(F) and P(T) are the prior probabilities of selecting the fair coin and the two-headed coin, respectively. Since the gambler selects one of the coins at random, the prior probabilities are equal:

P(F) = P(T) = 1/2.

The likelihoods of the evidence given the events can be calculated as follows:

P(H|F) = 1/2 (the fair coin is a fair coin)

P(H|T) = 1 (the two-headed coin always shows heads)

Substituting the values, we get:

P(F|H) = (1/2 * 1/2) / ((1/2 * 1/2) + (1 * 1/2)) = 1/3

Therefore, the probability that the gambler selected the fair coin given that the coin shows heads is 1/3.