Patterns and Algebra
The blue dots represent boys and the pink dots represent girls. This chart shows all possible birth orders in a family with two children. When we have an element with two outcomes that each have a 50% chance of happening and repeat it twice, each possible combination ends up having a 25% chance. The odds of two boys is 25%, the odds of two girls is 25%.
What are the odds of one of each? Well the number of possibilities where that happens is 2 out 4, so 50%. However, if we ask for them in a specific order, like girl then boy, the odds are 25%.
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How many possibilities are there where the first child is a boy? 2/4. 50%. What are the chances of the second child being a girl? There are again, two times out of four that happens. 50%.
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There is a 25% of both children being girls. If the first child is a girl, what are the chances the second one will be? It is tempting to say 25%, since that is the chance of two girls. However, the gender of the first child does not affect the odds of the second one. If we look at how many combinations have the second child as a girl, it is 50%. If we look at just the ones where the first child is a girl, still 1 in 2 has the second child being a girl. 50%.
First Child: Boy
First Child: Girl
Second Child: Girl
Second Child: Boy
First Child: Boy
Third Child: Girl
Third Child: Boy
First Child: Girl
This chart shows all possible birth order combinations for a family with three children. As you can see, I took the four options in the above chart, and then subdivided each quadrant into two options based the gender of a third child. This creates a total of eight options. Each birth order has a 1 in 8 chance, which is 12.5%. What are the odds of three children all being boys? 12.5%. So when the first two are boys, what are the odds of the third being a boy? It is tempting to say 12.5% because that is the overall odds of three boys, but again the odds of the third being a boy is independent of the first two. If you look at all possible combinations, how many have the third child male? 4 out of 8, which is 50%. If we look just at the combinations where the first two are boys, again 1 out of 2 has the third child as a boy, 50%.
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What are the chances of two girls and a boy? Count the number of combinations with that. 3 out of 8. 37.5%. The odds of three girls is less than the odds of two girls and a boy, however the odds of two girls and a boy in a specific order would be the same as the odds of three girls. GGB is the same odds as GGG or GBG, etc... All possible orders are the same odds as one another, but if we don't care about the specific order than some combinations become more likely than others.
Second Child: Girl
Second Child: Boy
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Let's imagine what would happen if we kept adding more children to the graph. Pictured here is one of the eight possible outcomes we identified with three. As you can see, a fourth element would split it into two more possibilities. Each possibility would be split into two new ones. Each new possibility would have a 1 in 16 chance of happening. to calculate the percentage, take the percentage that represents 1/8 (12.5%) and cut in half. 1/16=6.25%. If we continue this pattern (see how patterning and probability relate?) we see that adding a fifth element gives us 32 possibilities each with a 3.125% chance. A sixth element would give us 64 possibilities each with a 1.625% chance.
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To simplify that last number, let's say <2% (less than 2%). If a woman has six grandchildren, the odds of al six being girls is <2%. It seems like extremely low odds, until you consider every possible order is the same odds. If we eliminate specific order as a variable, there are more chances of it being five girls and a boy. Still the chances of girls and then a boy is just as unlikely as six girls. After five girls, it is tempting to say the odds of a sixth is <2% because that is the overall odds of six girls. However, by that time you have already narrowed the possibilities to two. It will be six girls or it will be five girls and then one boy. Intuitively, we think each girl lowers the odds of the next being a girl, but the odds for each element is independent of the other elements. Out of 64 possibilities, 32 have the sixth child as a girl. That is 50% no matter what gender the other five were.
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The same logic applies to any sequence where each element has approximately 50/50 odds. When flipping a coin, the odds of six heads in a row is <2%, but so is any possible combination of six tosses. The next toss, contrary to our intuition, still has a 50% chance of being heads and a 50% of being tails.