This problem involves conditional probability and the use of Bayes' Theorem. Let's solve it step by step:

Problem Breakdown

• Two coins in the pocket:

  1. A fair coin (1 head and 1 tail).
  2. A two-headed coin (2 heads).

• Experiment: A coin is selected at random and flipped twice.

• Outcome: Two heads are observed.

• Question: What is the probability that the selected coin was the two-headed coin?

Solution

Let:

• $C_1$: The event that the fair coin is selected.
• $C_2$: The event that the two-headed coin is selected.
• $E$: The event of flipping two heads.

Step 1: Assign Prior Probabilities

Since the coin is selected randomly, the probability of choosing either coin is: $P(C_1) = P(C_2) = \frac{1}{2}.$

Step 2: Compute Likelihoods

• If the fair coin ($C_1$) is selected:

  • Probability of flipping two heads: $P(E \mid C_1) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}.$

• If the two-headed coin ($C_2$) is selected:

  • Probability of flipping two heads: $P(E \mid C_2) = 1.$

Step 3: Use Bayes' Theorem

We calculate $P(C_2 \mid E)$, the probability that the selected coin is the two-headed coin given that two heads are observed: $P(C_2 \mid E) = \frac{P(E \mid C_2) \cdot P(C_2)}{P(E)}.$

Step 4: Compute $P(E)$

The total probability of flipping two heads ($P(E)$) is: $P(E) = P(E \mid C_1) \cdot P(C_1) + P(E \mid C_2) \cdot P(C_2).$ Substitute the values: $P(E) = \left( \frac{1}{4} \cdot \frac{1}{2} \right) + \left( 1 \cdot \frac{1}{2} \right) = \frac{1}{8} + \frac{4}{8} = \frac{5}{8}.$

Step 5: Compute $P(C_2 \mid E)$

Now substitute everything into Bayes' Theorem: $P(C_2 \mid E) = \frac{P(E \mid C_2) \cdot P(C_2)}{P(E)} = \frac{1 \cdot \frac{1}{2}}{\frac{5}{8}} = \frac{\frac{1}{2}}{\frac{5}{8}} = \frac{4}{5}.$

Final Answer:

The probability that the gambler selected the two-headed coin is: $\boxed{\frac{4}{5}}$

Would you like a detailed explanation of any step?

Relative Questions:

1. How would the probabilities change if the coin was flipped three times instead of two?
2. What is the probability of selecting the fair coin given the same conditions?
3. Can you derive the formula for Bayes' Theorem from scratch?
4. What happens to the result if there were three coins with different probabilities?
5. How can conditional probability be applied to real-world situations like medical testing?

Tip:

Bayes' Theorem is most useful when you want to reverse conditional probabilities, especially when dealing with observational data.