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Two coins are avilable, one fair and the other two-headed. Choose a coin and toss it once; assume that the unbiased coin is chosen with probalility 3/4. Given that the outcome is head, the probability that the two-headed coin waschosen is
- a)3/5
- b)2/5
- c)1/5
- d)2/7
Correct answer is option 'B'. Can you explain this answer?
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Two coins are avilable, one fair and the other two-headed. Choose a co...
B is the event of selecting a biased coin
UB is the event of selecting an unbiased coin
UB is the event of selecting an unbiased coin
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Two coins are avilable, one fair and the other two-headed. Choose a co...
Given information:
- Two coins are available - one fair and the other two-headed.
- Probability of choosing the unbiased coin is 3/4.
- We toss the chosen coin once.
- The outcome is a head.
To find:
- Probability that the two-headed coin was chosen.
Solution:
Let's use Bayes' theorem to solve this problem. Bayes' theorem states that the probability of an event A given event B is equal to the probability of event B given event A multiplied by the probability of event A, divided by the probability of event B.
Let's define the events:
- A: The two-headed coin was chosen.
- B: The outcome is a head.
We need to find P(A|B), the probability that the two-headed coin was chosen given that the outcome is a head.
Using Bayes' theorem, we have:
P(A|B) = P(B|A) * P(A) / P(B)
We know that P(B|A) = 1, since if we choose the two-headed coin, we are guaranteed to get a head. We also know that P(A) = 1/4, since we have a 1/4 chance of choosing the two-headed coin.
Now we need to find P(B), the probability of getting a head. We can use the law of total probability to find this:
P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)
We already know P(B|A) and P(A). Let's find P(B|not A) and P(not A):
- P(B|not A) = 1/2, since the fair coin has a 1/2 chance of landing on heads.
- P(not A) = 3/4, since we have a 3/4 chance of choosing the fair coin.
Now we can plug in all the values:
P(B) = 1 * 1/4 + 1/2 * 3/4 = 5/8
P(A|B) = 1 * 1/4 / (5/8) = 2/5
Therefore, the probability that the two-headed coin was chosen given that the outcome is a head is 2/5, which is option (B).
- Two coins are available - one fair and the other two-headed.
- Probability of choosing the unbiased coin is 3/4.
- We toss the chosen coin once.
- The outcome is a head.
To find:
- Probability that the two-headed coin was chosen.
Solution:
Let's use Bayes' theorem to solve this problem. Bayes' theorem states that the probability of an event A given event B is equal to the probability of event B given event A multiplied by the probability of event A, divided by the probability of event B.
Let's define the events:
- A: The two-headed coin was chosen.
- B: The outcome is a head.
We need to find P(A|B), the probability that the two-headed coin was chosen given that the outcome is a head.
Using Bayes' theorem, we have:
P(A|B) = P(B|A) * P(A) / P(B)
We know that P(B|A) = 1, since if we choose the two-headed coin, we are guaranteed to get a head. We also know that P(A) = 1/4, since we have a 1/4 chance of choosing the two-headed coin.
Now we need to find P(B), the probability of getting a head. We can use the law of total probability to find this:
P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)
We already know P(B|A) and P(A). Let's find P(B|not A) and P(not A):
- P(B|not A) = 1/2, since the fair coin has a 1/2 chance of landing on heads.
- P(not A) = 3/4, since we have a 3/4 chance of choosing the fair coin.
Now we can plug in all the values:
P(B) = 1 * 1/4 + 1/2 * 3/4 = 5/8
P(A|B) = 1 * 1/4 / (5/8) = 2/5
Therefore, the probability that the two-headed coin was chosen given that the outcome is a head is 2/5, which is option (B).
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Two coins are avilable, one fair and the other two-headed. Choose a coin and toss it once; assume that the unbiased coin is chosen with probalility 3/4. Given that the outcome is head, the probability that the two-headed coin waschosen isa)3/5b)2/5c)1/5d)2/7Correct answer is option 'B'. Can you explain this answer?
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