Bayesian Analysis of Coin Flipping: Fair and Trick Coins

Imagine you have a bag containing two coins. One is a fair coin with heads and tails, while the other is a trick coin that always shows heads. Without looking, you randomly draw one coin and flip it, and it comes up heads. What is the probability that the coin you chose was the fair coin? This problem can be solved using Bayesian analysis.

Introduction to the Problem

Let's define the events:

F: the event that the chosen coin is the fair coin. T: the event that the chosen coin is the trick coin (with two heads). H: the event that the flip results in heads.

We need to find the probability that the coin is the fair coin given that we flipped heads, i.e., P(F|H).

Step-by-Step Bayesian Analysis

Step 1: Calculate the Prior Probabilities

Since there are two coins, the prior probabilities are:

P(F) 1/2: the probability of choosing the fair coin. P(T) 1/2: the probability of choosing the trick coin.

Step 2: Calculate the Likelihoods

Next, we calculate the probability of flipping heads for each coin:

If the chosen coin is fair (F), the probability of heads is P(H|F) 1/2. If the chosen coin is trick (T), the probability of heads is P(H|T) 1 since it has heads on both sides.

Step 3: Calculate the Total Probability of Heads

Using the law of total probability, we can find P(H):

[ P(H) P(H|F) cdot P(F) P(H|T) cdot P(T) ]

Substituting the values we calculated:

[ P(H) left(frac{1}{2} cdot frac{1}{2}right) (1 cdot frac{1}{2}) frac{1}{4} frac{1}{2} frac{3}{4} ]

Step 4: Apply Bayes Theorem

Now we apply Bayes theorem to find the probability that the coin is the fair coin given that it came up heads:

[ P(F|H) frac{P(H|F) cdot P(F)}{P(H)} ]

Substituting the known values:

[ P(F|H) frac{left(frac{1}{2}right) cdot left(frac{1}{2}right)}{frac{3}{4}} frac{frac{1}{4}}{frac{3}{4}} frac{1}{3} ]

Conclusion

The probability that the coin you chose was the fair coin given that it came up heads is 1/3.

Frequentist vs. Bayesian Approach

Some might argue that since there are two coins, the odds of choosing either coin is and will always be 50/50 if you choose randomly. However, the Bayesian approach allows us to update our beliefs based on new evidence, in this case, the outcome of the coin flip.

The Bayesian analysis provides a more nuanced understanding of the situation, showing that the evidence from the coin flip significantly changes the probability from 50/50 to 1/3 in favor of the fair coin.

Further Insights

Understanding these concepts can be crucial in various fields, including data science, machine learning, and economics. The ability to update probabilities based on new evidence is a core principle in Bayesian statistics.