Boy or Girl Paradox if You Dont Know How Large the Family Is
Earlier today I set you lot the following four questions:
1. Mrs Smith has two children. The eldest one is a boy. What'southward the chance that both are boys.
2. Mrs Jones has two children. At least one is a boy. What's the chance that both are boys?
iii. Mrs Robinson has 2 children. At least one is a boy built-in on a Monday. What's the run a risk that both are boys?
iv. Mrs Taylor has two children. At least ane is a male child called Oscar. What'due south the chance that both are boys?
(Assume Mrs Smith, Jones, Robinson and Taylor are each chosen randomly from the population of families with exactly two children. The phrase 'at least 1 is a male child' is understood in the literal sense, i.e. in this case either one kid is a boy, or both children are boys.)
The answers are: 1. 1/2 or 50 per cent, 2. 1/three or 33.3 per cent, three. 13/27 or virtually 48 per cent, 4. very very close to ane/2 or fifty per cent
Yes, the answers are all different. It does make a difference when y'all mention the child'due south twenty-four hour period of birth, or proper noun.
We'll become to the workings after. First, the results of the questionnaire I asked parents of exactly 2 children to fill in.
one. 2 child families where eldest child is a male child. Full submissions: 4949. Of these families, the percentage with two boys was 44.9 per cent.
ii. Two child families where at to the lowest degree i child is a boy. Total submissions: 6060. Of these families, the percentage with two boys was 39.5 per cent.
3. Two child families where at to the lowest degree one kid is a boy born on a Monday. Of these families, the percentage with 2 boys was 54.0 per cent.
4. Two child families where eldest child is a boy named Oscar. Full submissions: 1961. Of these families, the percent with two boys was 55.half-dozen per cent.
In an ideal earth, in which two-parent families are selected genuinely randomly, in large enough numbers and respond truthfully, we'd expect the percentages of 'families with two boys' to judge the answers to the questions.
And they sort of do. [At that place is a large margin of error, merely, pleasingly, 2 is closest to 33 per cent, and the others all hover effectually 50 per cent. A surprising percentage of full respondents have sons called Oscar, although I guess I did, shamelessly and unscientifically, propose you could submit for friends who had a son called Oscar.]
Most people notice it hard to believe that the probabilities change when y'all mention the week day or name. I hope the questionnaire might give a petty insight every bit to why it might be the case.
Allow'south now get to the nitty gritty. In the proofs below, we are assuming that the take a chance of a boy or a girl is 50/50. And also that the chance of being born on any twenty-four hour period of the week is equal. (Neither of these might exist truthful, equally readers have said BTL.)
1. Mrs Smith has 2 children. The eldest i is a boy. What's the chance that both are boys.
If the eldest one is a boy, the youngest ane is either a boy or a girl, with a 50/fifty chance of each. And so the hazard both are boys is 1 in 2, or 50 per cent.
2. Mrs Jones has ii children. At least one is a boy. What'south the chance that both are boys?
If at least one is a boy, there are three possible equally probable gender-assignations of 2 siblings. male child-male child, boy-girl, or girl-boy. Merely 1 in 3 cases, or 33 per cent are both boys. The lesson here is that when because equally likely scenarios we must consider birth lodge. If the birth order of the boy is not specified – i.eastward. if we don't know if he is the eldest or the youngest – the probability of 2 boys drops to ane in three.
3. Mrs Robinson has two children. At to the lowest degree one is a male child born on a Monday. What'south the gamble that both are boys?
There are many ways to solve this. I think the easiest to empathize is by considering frequencies. Imagine selecting 196 two-children families at random. (It will later get clear why I chose this number).
We'd expect the post-obit frequencies of the four possible pairs of siblings, BB, BG, GB, GG, because each of the four sets is equally likely. (B is male child, One thousand is girl. Eldest child is kickoff one, youngest kid is second i)
-
BG 49
-
GB 49
-
BB 49
-
GG 49
At present let BM exist a boy built-in on a Monday, and BNM be a boy non born on a Mon. Our expected frequencies become
-
BK1000 seven
-
BNMChiliad 42
-
GBOne thousand 7
-
GBNOne thousand 42
-
B1000BM 1
-
BNMBM vi
-
BThousandBNM half-dozen
-
BNMBNM 36
-
GG 49
The number of families with a boy born on a Monday is seven + seven + i + six + 6 = 27. Of these, thirteen have two boys, and so the probability of having two boys is 13/27, which is about 48 per cent.
4. Mrs Taylor has ii children. At least 1 is a boy chosen Oscar. What's the chance that both are boys?
Here the unknown is the chance of a boy being called Oscar. I'd say, even among Guardian readers, it is going to be quite low. Say, 1 in 500 boys. We now do the same calculation every bit in part three only instead of a 1/seven chance of a particular twenty-four hour period, there is a 1/500 of a particular name. The result is going to be very close merely not equal to ane in 2. (If anyone wants to fill in the dots of the calculation below, please practice.)
To summarise, when no information is specified apart from the child'south gender (function 2) the chances of two boys is 33 per cent. The more than information that is supplied nigh the child – such as day of birth (part three), and name (role four) – the more than the chances of two boys approaches 50 per cent. When nascency gild is specified (office 1), the chances are exactly fifty per cent.
Some mathematicians honey this problem, considering the reply is so counter-intuitive. Others don't like it, because they feel the linguistic communication is misleading, or that at that place is no realistic situation when anyone would ever meaningfully state that 'at least 1' of their children is a male child. Rob Eastaway explores that question in his blogpost here.
I hope you lot enjoyed today's puzzle. I'll exist back in two weeks.
If y'all need a puzzle fix before then, my new volume And so You Think You've Got Problems? is just out. It'southward a compendium of 200 or so brainteasers with the stories behind them. 1 of the problems is the one discussed in today's column, with added material well-nigh the history and development of probability 'paradoxes' from the nineteenth century to the nowadays day. The other puzzles bridge wordplay, logic, geometry, linguistics, topology and many other fields. It's for all abilities: there are very simple teasers that children will be able to do and utterly inexplainable ones for just the sharpest minds. Would brand a great Christmas souvenir!
I fix a puzzle hither every two weeks on a Monday. I'm always on the look-out for not bad puzzles. If you would like to suggest i, email me.
Source: https://www.theguardian.com/science/2019/nov/18/did-you-solve-it-the-two-child-problem
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