Expected Size of One-Boy Family and Probability of Girls

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The expected size of a one-boy family can be calculated using the formula E[X] = α + (1 - α) × x, where α represents the probability of a random birth being a boy. For α = 0.51, the expected size of a one-boy family is 1.49. This indicates that, on average, a one-boy family will have approximately 1.49 children, factoring in the likelihood of having girls. The confusion arises from the terminology, as a one-boy family refers to a family that has at least one boy, not exclusively boys.

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X is the number of girls in a one-boy family, and so X+1=the size of a one-boy family. What is the expected size of a one-boy family in terms of [tex]\alpha[/tex]? What is the expected size of a one-boy family if [tex]\alpha[/tex]=0.51?

[tex]\alpha[/tex]=the probability that a random birth is a boy

I know this isn't difficult, but the terminology has completely gotten me confused. How do I solve for this with an actual formula?
 
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Would this be equivalent to:

[tex]E[X]=\alpha + (1- \alpha) \times x[/tex]
 
What is a one-boy family? How can there be girls in a one-boy family?
 

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