A couple has two children. What is the probability that both are boys if it is known that one of them is a boy
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Sample space: S = {MM, MF, FM, FF}, where M = male and F = female.
(i) To find P (both children are males, if it is known that at least one of the children is male).
A: Event that both children are male, and B: event that at least one of them is a male.
A: {MM} and B: {MF, FM, MM} →→ P (A ∩∩ B) = {MM}
Probability that both are males, if we know one is a female = P(A/B)=[P(A∩B)/P(B)]
Given S = {MM, MF, FM, FF}, we can see that: P (A) = 1/4; P (B) = 3/4; P (A ∩B) = 1/4
Therefore, P(B/A)=P(BA)=[P(A∩B)/P(A)]=1/4×2/1= 1/2
(i) To find P (both children are males, if it is known that at least one of the children is male).
A: Event that both children are male, and B: event that at least one of them is a male.
A: {MM} and B: {MF, FM, MM} →→ P (A ∩∩ B) = {MM}
Probability that both are males, if we know one is a female = P(A/B)=[P(A∩B)/P(B)]
Given S = {MM, MF, FM, FF}, we can see that: P (A) = 1/4; P (B) = 3/4; P (A ∩B) = 1/4
Therefore, P(B/A)=P(BA)=[P(A∩B)/P(A)]=1/4×2/1= 1/2
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