Understanding the Probability Problem

This problem asks us to calculate the probability of a specific event (selecting the two-headed coin) given that another event has already occurred (the coin flip resulted in heads). This type of problem involves conditional probability and is typically solved using Bayes' Theorem.

Setting Up the Problem: Coins and Probabilities

We have a box containing three distinct coins:

• Coin 1 (\(C_1\)): A two-headed coin.
• Coin 2 (\(C_2\)): A fair coin.
• Coin 3 (\(C_3\)): A biased coin.

Since one coin is selected at random from the three, the initial probability of selecting any specific coin is equal:

• Probability of selecting the two-headed coin, \(P(C_1) = \frac{1}{3}\).
• Probability of selecting the fair coin, \(P(C_2) = \frac{1}{3}\).
• Probability of selecting the biased coin, \(P(C_3) = \frac{1}{3}\).

Let \(H\) be the event that the coin flip results in heads. The probability of getting heads depends on the type of coin selected:

• If \(C_1\) (two-headed) is flipped, the probability of heads is \(P(H|C_1) = 1\) (or 100%).
• If \(C_2\) (fair) is flipped, the probability of heads is \(P(H|C_2) = 0.5\) (or 50%).
• If \(C_3\) (biased) is flipped, the probability of heads is \(P(H|C_3) = 0.75\) (or 75%).

We are given that the flipped coin shows heads, and we want to find the probability that the selected coin was the two-headed one, which is \(P(C_1|H)\).

Calculating the Total Probability of Getting Heads

Before we can find \(P(C_1|H)\) using Bayes' Theorem, we need to calculate the total probability of getting heads, \(P(H)\). We can use the Law of Total Probability for this. The Law of Total Probability states that if \(C_1, C_2, C_3\) are mutually exclusive and exhaustive events (they cover all possibilities), then \(P(H)\) can be calculated as:

\(P(H) = P(H|C_1)P(C_1) + P(H|C_2)P(C_2) + P(H|C_3)P(C_3)\)

Let's plug in the values we have:

\(P(H) = (1 \times \frac{1}{3}) + (0.5 \times \frac{1}{3}) + (0.75 \times \frac{1}{3})\)

\(P(H) = \frac{1}{3} + \frac{0.5}{3} + \frac{0.75}{3}\)

\(P(H) = \frac{1 + 0.5 + 0.75}{3}\)

\(P(H) = \frac{2.25}{3}\)

To simplify the fraction, we can write 2.25 as \(\frac{225}{100}\) or \(\frac{9}{4}\):

\(P(H) = \frac{\frac{9}{4}}{3} = \frac{9}{4 \times 3} = \frac{9}{12} = \frac{3}{4}\)

So, the total probability of getting heads is \(\frac{3}{4}\).

Applying Bayes' Theorem to Find the Conditional Probability

Now we can use Bayes' Theorem to find the probability that the coin selected was the two-headed coin (\(C_1\)) given that the flip resulted in heads (\(H\)). Bayes' Theorem is stated as:

\(P(C_1|H) = \frac{P(H|C_1) \times P(C_1)}{P(H)}\)

We have the values for each part of the formula:

• \(P(H|C_1) = 1\)
• \(P(C_1) = \frac{1}{3}\)
• \(P(H) = \frac{3}{4}\)

Substitute these values into the formula:

\(P(C_1|H) = \frac{1 \times \frac{1}{3}}{\frac{3}{4}}\)

\(P(C_1|H) = \frac{\frac{1}{3}}{\frac{3}{4}}\)

To divide by a fraction, we multiply by its reciprocal:

\(P(C_1|H) = \frac{1}{3} \times \frac{4}{3}\)

\(P(C_1|H) = \frac{1 \times 4}{3 \times 3}\)

\(P(C_1|H) = \frac{4}{9}\)

Thus, the probability that it was the two-headed coin, given that it showed heads, is \(\frac{4}{9}\).

Summary of Probabilities

Using Bayes' Theorem:

\(P(C_1|H) = \frac{P(H|C_1)P(C_1)}{P(H)} = \frac{1 \times \frac{1}{3}}{\frac{3}{4}} = \frac{\frac{1}{3}}{\frac{3}{4}} = \frac{1}{3} \times \frac{4}{3} = \frac{4}{9}\)

Revision Table: Key Probability Concepts

Additional Information: Understanding Bayes' Theorem

Bayes' Theorem is a fundamental concept in probability theory and statistics. It describes how to update the probability of a hypothesis based on new evidence. In this problem, the 'hypothesis' is which coin was selected (\(C_1\), \(C_2\), or \(C_3\)), and the 'evidence' is that the coin flip resulted in heads (\(H\)).

The term \(P(C_i)\) is the prior probability of selecting coin \(C_i\) before any flip happens. The term \(P(H|C_i)\) is the likelihood, the probability of observing the evidence (heads) given a specific hypothesis (coin type). The term \(P(H)\) is the marginal likelihood or total probability of the evidence. Finally, \(P(C_i|H)\) is the posterior probability, the updated probability of selecting coin \(C_i\) after observing that the flip was heads.

Bayes' Theorem allows us to move from knowing the probability of evidence given a hypothesis (\(P(H|C_i)\)) to knowing the probability of the hypothesis given the evidence (\(P(C_i|H)\)), which is often what we are interested in in real-world scenarios.