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Consider the experiment of throwing a dice, if a multiple of '3' comes up, throw the dice again and if any other number comes, toss a coin. Find the conditional probability of the event coin shows the tail, given that at least one die shows a '3'.
- a)0
- b)1
- c)2
- d)3
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Consider the experiment of throwing a dice, if a multiple of 3 comes u...
The sample space of the experiment is,
Let A be the event that the coin shows A tail and B be the event that at least one die shows 3.
A = {(1, T), (2, T), (4, T), (5, T)}
B = {(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (6, 3)}
⇒ A ∩ B = φ ∴ P (A ∩ B) = 0
(B)= P{(3, 1)} + {(3, 2)} + {(3, 3)} + {(3, 4)} + {(3, 5)} + {(3, 6)} +[(6,3)
Probability of the event that the coin shows a tail, given that at least one die shows 3, is given by P(A | B).
∴
Let A be the event that the coin shows A tail and B be the event that at least one die shows 3.
A = {(1, T), (2, T), (4, T), (5, T)}
B = {(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (6, 3)}
⇒ A ∩ B = φ ∴ P (A ∩ B) = 0
(B)= P{(3, 1)} + {(3, 2)} + {(3, 3)} + {(3, 4)} + {(3, 5)} + {(3, 6)} +[(6,3)
Probability of the event that the coin shows a tail, given that at least one die shows 3, is given by P(A | B).
∴
Most Upvoted Answer
Consider the experiment of throwing a dice, if a multiple of 3 comes u...
Understanding the Experiment
In this experiment, we have two main actions based on the outcome of a dice throw:
- If a multiple of 3 (i.e., 3 or 6) is rolled, we throw the dice again.
- If any other number (1, 2, 4, 5) is rolled, we toss a coin.
Identifying Outcomes
- The possible outcomes of rolling a die are: 1, 2, 3, 4, 5, and 6.
- The multiples of 3 are: 3 and 6.
- The non-multiples of 3 are: 1, 2, 4, and 5.
Condition of Interest
We need to find the conditional probability that the coin shows tails given that at least one die shows a 3.
Events Analysis
1. Rolling a 3:
- If the first roll is a 3, we roll again.
- At this point, possibilities are again 1, 2, 3, 4, 5, or 6.
2. Rolling again after a 3:
- If the second roll is 3 again, we repeat the process, and this could continue indefinitely.
- If we roll a non-multiple (1, 2, 4, or 5), we toss a coin.
3. Rolling a 6:
- If the first roll is a 6, we roll again, maintaining the same analysis as before.
Conditioning on the Event
Given that at least one die shows a 3, it is impossible for the coin to show tails because the only time we toss a coin is when a non-multiple of 3 is rolled. Since we rolled a 3 first, the only possibilities afterward involve rolling again, which may not yield a non-multiple.
Conclusion
Thus, the probability of the coin showing tails, given that at least one die shows a 3, is 0.
Therefore, the correct answer is:
Answer: 0 (Option A)
In this experiment, we have two main actions based on the outcome of a dice throw:
- If a multiple of 3 (i.e., 3 or 6) is rolled, we throw the dice again.
- If any other number (1, 2, 4, 5) is rolled, we toss a coin.
Identifying Outcomes
- The possible outcomes of rolling a die are: 1, 2, 3, 4, 5, and 6.
- The multiples of 3 are: 3 and 6.
- The non-multiples of 3 are: 1, 2, 4, and 5.
Condition of Interest
We need to find the conditional probability that the coin shows tails given that at least one die shows a 3.
Events Analysis
1. Rolling a 3:
- If the first roll is a 3, we roll again.
- At this point, possibilities are again 1, 2, 3, 4, 5, or 6.
2. Rolling again after a 3:
- If the second roll is 3 again, we repeat the process, and this could continue indefinitely.
- If we roll a non-multiple (1, 2, 4, or 5), we toss a coin.
3. Rolling a 6:
- If the first roll is a 6, we roll again, maintaining the same analysis as before.
Conditioning on the Event
Given that at least one die shows a 3, it is impossible for the coin to show tails because the only time we toss a coin is when a non-multiple of 3 is rolled. Since we rolled a 3 first, the only possibilities afterward involve rolling again, which may not yield a non-multiple.
Conclusion
Thus, the probability of the coin showing tails, given that at least one die shows a 3, is 0.
Therefore, the correct answer is:
Answer: 0 (Option A)
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