Suppose a married couple has four children. What is the probability that two of them are girls and two are boys? Assuming that half of births are male and half female, common sense leads us to believe that the family will have two boys and two girls. But it can be mathematically shown that this is quite unlikely. Welcome to the paradox of the four children!
Our brain tends to play tricks on us when it assumes results based on what people call 'common sense'. When we compare the results obtained by this intuitive method with those yielded by cold (but effective) mathematical calculations, we are surprised to see how wrong we were. One of the paradoxes that is easiest to demonstrate is the one that Martin Gardner - an American science popularizer and philosopher of science - calls the “paradox of the four children”.
Gardner says that if we know (or are told) that a couple has four children, we tend to think that there is a high probability that two of them will be boys and two girls. However, despite the fact that statistically almost exactly half of births are male and half female, it can be mathematically shown that our intuition fails miserably.
The way to approach this problem is really simple. Suppose we represent each birth of a boy with an H (for hombre, man) and that of a girl with an M (for mujer, woman). We only have to draw up what is called a truth table representing all the different possibilities when having four children. In the following table, the order from left to right indicates the order of birth:
The 16 possible combinations
- HHHH
- HHHM
- HHMH
- HHMM
- HMHH
- HMHM
- HMMH
- HMMM
- MHHH
- MHHM
- MHMH
- MHMM
- MMHH
- MMHM
- MMMH
- MMMM
Since there are only two possible sexes, the number of combinations for four births is the 16 shown in the table above. Remember that our entire analysis is valid because we are considering that the probability of being a boy is equal to that of being a girl (50% each). In the real world, this proportion is not exact, but it is close enough that the results we are going to show practically do not vary.
The next step is to count each of the cases shown in the table. We see that, of the 16, there are only two cases in which the sex of all the children is the same (1 and 16). That means we have a probability of 2/16 (or 1/8, or 12.5%) that our four children are all the same sex. If we count the cases where births include one child of one sex and three of the other, we find eight cases (rows 2,3,5,8,9,12,14, and 15). This implies that in half of the cases, a couple that has 4 children will have either one girl and three boys, or one boy and three girls.
Finally, if we count the cases that interest us, those in which there are two children of each sex, we see that only cases 4, 6, 7, 10, 11, and 13 meet the condition of “two boys and two girls.” This shows that only 6 out of 16 times (or 3 out of 8, if we 'simplify') does the situation that our common sense said was the most likely occur. Mathematics shows that only 37.5% of families with four children will have two of each sex, and that in reality it is much more likely to have three children of one sex and one of the other than any of the other possibilities individually.
This result disconcerts us because something in our mind makes us relate the fact that the probability of having a son or daughter is 50% with the erroneous conclusion that the most logical thing is to have the same number of boys as girls. But that is valid only if we have two children. With four - as we have seen - the possibilities are reduced, showing that we cannot always trust our common sense. What do you think?
