Please help me with this calculus based statistics question which involves findi

Question

Please help me with this calculus based statistics question which involves finding the probabilities and expected value. Please be DETAILED and show all work. If not much work is shown, I will rate the answer poorly.

Thank you

4. (a) A couple decide to have children until they have a daughter. What is the probability of having less than 3 children? (b) What is the expected number of their children? (c) Another couple decide to have children until they have two sons. What is the probability of 5 children?

Explanation / Answer

a)Probabilty (daughter)=0.5

P(childen <3)= P(children =1)+ P(children= 2)

                  =0.5 +0.5*0.5

                   =0.75

b)Expected no children =1*0.5 +2* 0.52 +3*0.53 +........................infinity

                                =0.5*(1/0.52 ) =2

c)Probabilty for 5 Children =(Probality for 1 out of first 4 children is son and rest are daughter)*((Probality for 5th child to be son) =(4*0.5*0.53) *0.5

                        =0.125

                                 

                                    

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