If the probability of having n children in a family is: PN = α p ^ n, let the birth rates of boys and girls be equal, find the probability of having K boys in the family? In addition, if there is at least one boy in the family, find the probability of having at least 2 boys in the family; if there is no girl in the family, find the probability of having exactly one boy

If the probability of having n children in a family is: PN = α p ^ n, let the birth rates of boys and girls be equal, find the probability of having K boys in the family? In addition, if there is at least one boy in the family, find the probability of having at least 2 boys in the family; if there is no girl in the family, find the probability of having exactly one boy

I can only write the summation formula for the first question of this question: suppose there are n children, among which X boys are born, then the probability is Nck * (1 / 2) ^ n * α p ^ n
Then sum the above formula from n = k to infinity, and get the probability that there are k boys in the family
The second problem, my method is a little complicated: first, find out the probability of at least 2 boys in a family, then just find out the probability of having 0 boys and 1 boy, and subtract it by 1. If there are n children, none of them is a boy, P = α p ^ n * (1 / 2) ^ n, sum from 0 to infinity... From P / 2 is necessarily less than 1, the limit is α / (1-p / 2), Similarly, find the probability of birth of a boy ∑ (n from 1 to infinity) α n (P / 2) ^ n
Then to find at least one boy is to divide 1-p (no boy) = 1 - α / (1-p / 2). Two probabilities are conditional probabilities
Third: the probability that there is no girl and just one boy in the family is α P / 2, and the probability that there is no girl is α / (1-p / 2). The conditional probability p / 2 - (P ^ 2) / 4 is obtained by dividing the two formulas