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A family has 2 children. The probability that both of them are boys if it is known that one of them is a boy
- a)1
- b)1/3
- c)3/4
- d)none
Correct answer is option 'B'. Can you explain this answer?
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Solution:
This problem can be solved by using Bayes' theorem.
Bayes' theorem states that the probability of an event (A) given that another event (B) has occurred is equal to the probability of both events occurring (A and B), divided by the probability of event B occurring.
P(A|B) = P(A and B) / P(B)
In this case, we want to find the probability that both children are boys (event A), given that we know one of them is a boy (event B).
Let's define the following events:
A: Both children are boys
B: One of the children is a boy
We know that event B has occurred, so we can write:
P(A|B) = P(A and B) / P(B)
P(B) is the probability that one of the children is a boy, which can be calculated as follows:
P(B) = 1 - P(both children are girls)
P(B) = 1 - 1/4 (since the probability of both children being girls is 1/4)
P(B) = 3/4
Now we need to calculate P(A and B), which is the probability that both children are boys and one of them is a boy. This is simply the probability that both children are boys, since we already know that one of them is a boy.
P(A and B) = P(both children are boys)
P(A and B) = 1/4
Finally, we can plug these values into Bayes' theorem to get:
P(A|B) = P(A and B) / P(B)
P(A|B) = (1/4) / (3/4)
P(A|B) = 1/3
Therefore, the probability that both children are boys, given that we know one of them is a boy, is 1/3.
This problem can be solved by using Bayes' theorem.
Bayes' theorem states that the probability of an event (A) given that another event (B) has occurred is equal to the probability of both events occurring (A and B), divided by the probability of event B occurring.
P(A|B) = P(A and B) / P(B)
In this case, we want to find the probability that both children are boys (event A), given that we know one of them is a boy (event B).
Let's define the following events:
A: Both children are boys
B: One of the children is a boy
We know that event B has occurred, so we can write:
P(A|B) = P(A and B) / P(B)
P(B) is the probability that one of the children is a boy, which can be calculated as follows:
P(B) = 1 - P(both children are girls)
P(B) = 1 - 1/4 (since the probability of both children being girls is 1/4)
P(B) = 3/4
Now we need to calculate P(A and B), which is the probability that both children are boys and one of them is a boy. This is simply the probability that both children are boys, since we already know that one of them is a boy.
P(A and B) = P(both children are boys)
P(A and B) = 1/4
Finally, we can plug these values into Bayes' theorem to get:
P(A|B) = P(A and B) / P(B)
P(A|B) = (1/4) / (3/4)
P(A|B) = 1/3
Therefore, the probability that both children are boys, given that we know one of them is a boy, is 1/3.
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A family has 2 children. The probability that both of them are boys if it is known that one of them is a boya)1b)1/3c)3/4d)noneCorrect answer is option 'B'. Can you explain this answer?
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