Solving the Coin Flip Probability Puzzle: Which Coin Showed Heads?

 Suppose you have ten coins, each with a unique probability of showing heads when flipped. If the i-th coin is flipped, it will show heads with probability i/10, where i ranges from 1 to 10. Now, imagine that one of these ten coins is randomly selected and flipped, showing heads. What is the probability that the fifth coin was the one that was flipped?

To solve this probability puzzle, we can use Bayes' theorem, which is a formula that calculates conditional probability. Conditional probability is the likelihood of an event occurring given that another event has already occurred. In this case, we want to know the probability of selecting the fifth coin given that it showed heads.

Let's define some variables to make this problem easier to solve:

F: The event that the fifth coin was selected

H: The event that the selected coin showed heads

We want to find the probability of F given H, or P(F|H). Bayes' theorem tells us that:

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

Here, P(H|F) is the probability of the selected coin showing heads given that the fifth coin was selected, P(F) is the probability of selecting the fifth coin, and P(H) is the probability of the selected coin showing heads (regardless of which coin it is).

Let's calculate each of these probabilities.

First, P(F) is simply 1/10, since there are ten coins and each has an equal chance of being selected.

Next, P(H) is the sum of the probabilities of each coin showing heads multiplied by their likelihood of being selected. This can be expressed as:

P(H) = Σ(i=1 to 10) P(H|i) * P(i)

where P(H|i) is the probability of the i-th coin showing heads, and P(i) is the probability of selecting the i-th coin. Substituting the given probability of each coin showing heads, we get:

P(H) = Σ(i=1 to 10) i/10 * 1/10

= 0.385

Finally, P(H|F) is simply the probability of the fifth coin showing heads, which is 5/10 or 0.5.

Now we can plug these values into Bayes' theorem:

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

= (0.5 * 1/10) / 0.385

≈ 0.1299

Therefore, the probability that the fifth coin was selected given that it showed heads is approximately 0.1299, or 12.99%.

This problem is an excellent example of conditional probability and how Bayes' theorem can be used to solve complex probability puzzles. Remember, the key to solving these problems is to break them down into smaller parts and carefully calculate each probability before plugging them into Bayes' theorem.