"It will be seen the the method of inverse probability is in complete accord with commonsense; indeed we could hardly expect it to be otherwise, once the axioms of probability have themselves been accepted as reasonable" , Philip Mayne Woodward

Introduction: In this article, we will explore an intriguing coin riddle and solve it using probability theory. The riddle goes as follows: There is a collection of coins, where 75% of the coins are genuine, and 25% are double-headed (dud) coins. A coin is randomly picked from the collection and tossed, resulting in a heads outcome. Given this information, what is the probability that the coin is genuine? We will solve this problem analytically using Bayes' theorem and provide a detailed explanation of the solution.

Problem Setup: Let's define some variables to represent the problem:

• Let A be a random variable representing the type of coin, where A = 1 for a dud coin and A = 0 for a genuine coin.
• Let R be a random variable representing the result of the coin toss, where R = 1 for heads and R = 0 for tails.

Given:

• P(A = 1) = 0.25 (the probability of picking a dud coin)
• P(A = 0) = 0.75 (the probability of picking a genuine coin)
• P(R = 1 | A = 1) = 1 (the probability of getting heads given a dud coin)
• P(R = 1 | A = 0) = 0.5 (the probability of getting heads given a genuine coin)

Step 1: Applying Bayes' Theorem To solve this problem, we will use Bayes' theorem, which allows us to calculate the probability of an event A given event B. The formula for Bayes' theorem is:

$$ P(A|R) = \frac{P(R|A) P(A)) } {P(R)} $$

In our case, we want to find P(A = 0 | R = 1), the probability that the coin is genuine given that it landed heads. We can rewrite Bayes' theorem as:

$$ P(A = 0 | R = 1) = \frac{P(R = 1 | A = 0) P(A = 0)}{P(R = 1)}, $$

Step 2: Calculating P(R = 1) To find P(R = 1), we can use the law of total probability:

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