What is the probability of third child of a couple?

Q: We have 3 children, they are all boys, and we now want another child. Should we give up our hope for a girl?

A: Even after 3 children of the same sex, your chances in the next pregnancy having the same one is about even. Having already 3 boys you may think it’s a foregone conclusion that the next baby is going to be another boy, but in reality, your chances are about equal between a boy and a girl. It’s not that unusual to have 3 children of the same gender in a row. This actually happens in one out of four couples who have 3 children.

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Even after 3 children of the same sex, your chances in the next pregnancy having the same one is about even.

How the baby’s gender is determined

Each of the man’s sperms and each egg carry 23 chromosomes. The sperm and the egg combine at fertilization for a total of 46 chromosomes which come in 23 pairs.
The 23rd chromosome is called the “sex-chromosome”. In sperms, it is either an X- or a Y-chromosome, while in the woman’s egg it is always X.
If an X-bearing sperm fertilizes the egg, the resulting 23rd pair is XX, a girl, and if a Y-bearing sperm fertilizes the egg, the result is XY, a boy.
It is therefore the sperm that decides the baby’s gender: X-bearing sperms result in girls, and Y-bearing sperms in boys.
In general, X- and Y-bearing sperms are about equally distributed in the ejaculate, so each time there is about an equal 50-50 chance for each gender.

If you already have a boy, the chance that the next baby is also a boy is 50 percent

It’s all a matter of mathematics. A couple having 4 children of the same gender in a row happens actually more often than you might think. In fact, one in eight couples (12.5%) with 4 children have 4 of the same sex.

Here's how it goes:

Couples with 2 children: 50% of couples have 2 girls or 2 boys, and 50% of couples have 1 girl/1 boy
Couples with 3 children: 25% of couples have 3 girls or 3 boys, and 75% have 1 boy/2 girls or 1 girl/2 boys
Couples with 4 children: 12.5% of couples have 4 girls or 4 boys, 37.5% will have 1/3 and 50% will have 2/2.
Couples with 5 children: 6.25% have 5 the same sex and in couples with 6 children, 3.125% will have the same sex.

	All same sex	Different sexes
2 children	50% G/B	50% B/G
3 children	25% all girl or all boys	75% BB/G or GG/B
4 children	12.5% all girls or all boys	37.5% 1B/3G or 1G/3B	50% BB/GG
5 children	6.25% all boys or all girls	25% BBBB/G or GGGG/B	68.75% GGG/BB or BBB/GG
6 children	3.125% all boys or all girls	12.5% BBBBB/G or GGGGG/B	46.875%: GGGG/BB or BBBB/GG

Read More:
How to Conceive a Boy or a Girl: The Shettles Method
Gender Predictor Test
Baby Eye Color Predictor

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Hint: The gender of the child is dependent on the father. Female gametes have only X chromosomes whereas the male gametes have a 50% chance of having X or Y

Complete answer:

If a couple has three children the chance of the individual child being a girl or a boy is 1/2 chance of the child being a girl and 1/2 the chance of the child being a boy. Thus if the couple were to have all-female children is 1/2 x 1/2 x 1/2 = 1/8.The gender of the child is determined by the chromosome present. A female has XX and a male has XY chromosomes present. When gametes are formed these gametes have one of the sex chromosomes from each parent. Since women have two X chromosomes the female gamete, the egg only has X. In male gametes since males have one X and one Y there will be a 50 % chance of the sperm carrying an X or a Y. When a sperm carrying X fuses with an egg the zygote will contain XX and hence the child will be a female. If the sperm carrying Y chromosome fuses with the egg, the zygote will have XY chromosome thus the child will be a male.

Note: The sex of the child is not related to other offsprings of the same parents, gender outcomes are independent and a matter of chance. If the sex of the children of successive births remains the same, the probability of that outcome for a single-sex increase.