A box contains the following three coins.
I. A fair coin with head on one face and tail on the other face.
II. A coin with heads on both the faces.
III. A coin with tails on both the faces.
A coin is picked randomly from the box and tossed. Out of the two remaining coins in the box, one coin is then picked randomly and tossed. If the first toss results in a head, the probability of getting a head in the second toss is
I. A fair coin with head on one face and tail on the other face.
II. A coin with heads on both the faces.
III. A coin with tails on both the faces.
A coin is picked randomly from the box and tossed. Out of the two remaining coins in the box, one coin is then picked randomly and tossed. If the first toss results in a head, the probability of getting a head in the second toss is
Show Hint
- \( \frac{2}{5} \)
- \( \frac{1}{3} \)
- \( \frac{1}{2} \)
- \( \frac{2}{3} \)
The Correct Option is B
Solution and Explanation
The first coin is chosen randomly from the three coins, and it is tossed. Given that the first toss results in a head, we need to calculate the probability of getting a head on the second toss. - The probability of selecting Coin I (fair coin) is \( \frac{1}{3} \), and if Coin I is selected, the probability of getting a head is \( \frac{1}{2} \). - The probability of selecting Coin II (double heads) is \( \frac{1}{3} \), and if Coin II is selected, the probability of getting a head is 1. - The probability of selecting Coin III (double tails) is \( \frac{1}{3} \), and if Coin III is selected, the probability of getting a head is 0. Step 2: Apply Bayes' Theorem.
Since the first toss results in a head, we can use Bayes' Theorem to update the probabilities of each coin being selected: \[ P(\text{Coin I} | \text{Head}) = \frac{P(\text{Head} | \text{Coin I}) P(\text{Coin I})}{P(\text{Head})} \] Similarly, we calculate for Coin II and Coin III. We then use the updated probabilities to find the probability of getting a head on the second toss, given the first toss was a head. After solving, we find the probability of getting a head on the second toss is \( \frac{1}{3} \). Thus, the correct answer is option (B). Final Answer: \( \frac{1}{3} \)
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