miscellaneous problem 2 on conditional probability

miscellaneous problem 2 on conditional probability


A couple has two children, Find the probability that both children are males, if it is known that at least one of the children is male.Also find the probability that both children are females, if it is known that the elder child is a female.

Let (x,y) denote the childre, where x stands for the elder child and y stands for the younger child.

Let M stand for male child and F stand for female child.

Sample space S = { (M,M) (M,F), (F,F),(F,M) }

let A denote the event that both children are males.
let B denote the event that at least one is a male.

A = {(M,M)}
B = {(M,M) (M,F),(F,M) }

interesection of A and  B, A ∩ B = {(M,M)}

P[ A ? B ] = (1 / 4)

P[B]  = (3 /4)

P [ both children are males given that at least one of the childre is a male ] = P [ A / B ]

                    P[ A ∩ B ]
P[ A / B ] = -------------
                       P[B] 

P[ A / B ] = (1/4) / (3 /4)

P[ A / B ] = ( 1/3 )

let E denote the event that both children are females.
let F denote the event that the elder child is a  female.

E = { (F,F) }
F = {(F,F),(F,M)}

interesection of E and  F, E ? F = {(F,F)}

P[E ∩ F] = ( 1/4 )
P[F] = ( 2/4 )

P[ both children are females given that the elder child is a female ] = P[ E/F ]


                   P[ E ∩ F ]
P[ E/F ] =  -------------------
                       P[F]


P[ E/F ] = ( 1/4 ) / ( 2/4 )  = ( 1/2 )

index of more problems on baye's theorem for ncert cbse xii mathematics
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