GATE2025_CS1_Q32

A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin is tossed, the probability P(head) = 0.5 and for a fake coin, P(head) = 1. You pick a coin at random and toss it twice, and get two heads. The probability that the coin you have chosen is the fake coin is ______. (rounded off to two decimal places)

✅ Final Answer:

The correct answer is 0.50.

🔹 Short Answer:

This is a conditional probability problem solved using Bayes' Theorem. We want to find P(Fake | 2 Heads). The probability of picking the fake coin and getting two heads is (1/5) * 1² = 0.2. The probability of picking a regular coin and getting two heads is (4/5) * 0.5² = 0.2. The total probability of getting two heads is 0.2 + 0.2 = 0.4. Therefore, the desired probability is (Probability of it being fake) / (Total probability) = 0.2 / 0.4 = 0.50.

🔸 Long Answer:

This problem asks for the probability of an event (choosing the fake coin) given that another event has occurred (getting two heads). This is a classic scenario for applying Bayes' Theorem.

Step 1: Define the Events

• Let F be the event that the chosen coin is Fake.
• Let R be the event that the chosen coin is Regular.
• Let HH be the event that tossing the coin twice results in two Heads.

We want to find the probability P(F | HH) - the probability it was the fake coin, given that we observed two heads.

Step 2: State Bayes' Theorem

The formula for Bayes' theorem in this context is:

P(F | HH) = [ P(HH | F) * P(F) ] / P(HH)

Step 3: Calculate Each Component

Prior Probabilities (before the experiment):

• P(F) = Probability of picking the fake coin = 1/5 = 0.2
• P(R) = Probability of picking a regular coin = 4/5 = 0.8

Likelihoods (probability of the evidence, given the cause):

• P(HH | F) = Probability of getting two heads if the coin is fake. Since P(Head|F) = 1, P(HH|F) = 1 * 1 = 1.0.
• P(HH | R) = Probability of getting two heads if the coin is regular. Since P(Head|R) = 0.5, P(HH|R) = 0.5 * 0.5 = 0.25.

Total Probability of the Evidence P(HH):
We use the Law of Total Probability. Two heads can occur in two mutually exclusive ways: we picked a fake coin and got HH, OR we picked a regular coin and got HH.

P(HH) = P(HH ∩ F) + P(HH ∩ R)
P(HH) = P(HH | F) * P(F) + P(HH | R) * P(R)

Plugging in the values:

P(HH) = (1.0 * 0.2) + (0.25 * 0.8)
P(HH) = 0.2 + 0.2 = 0.4

Step 4: Calculate the Final Posterior Probability

Now we substitute our calculated values back into the Bayes' formula:

P(F | HH) = [ P(HH | F) * P(F) ] / P(HH)
P(F | HH) = [ 1.0 * 0.2 ] / 0.4
P(F | HH) = 0.2 / 0.4 = 0.5

Step 5: Format the Answer

The question asks to round off to two decimal places. Therefore, 0.5 becomes 0.50.

Bibliography:
- Ross, S. M. (2014). A First Course in Probability. Pearson. (Chapter 3: Conditional Probability and Independence).
- Grinstead, C. M., & Snell, J. L. (2012). Introduction to Probability. American Mathematical Society. (Chapter 4: Conditional Probability).

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