Note: These are via @marcsmith (of unknown urlage) and so for once I don't know the answers either. I'll post my thoughts in the first comment
Quick
A country is preparing for a possible future war. Its tradition is to send only men into battle, so they want to increase the proportion of males to females in the population. A law is passed which requires every married couple to have children and to continue to do so until they have a boy. What effect do you expect this law to have on the balance of the population? (via BBC Radio 4 - More or Less)
Cryptic
There are an infinite number of even integers and an infinite number of odd integers. There are also hence an infinite number of integers overall. The number of even integers is the same as the number of odd integers. By how much does the set of all integers differ from the set of even integers? (via Georg Cantor)
Answer Update (after 8 comments)
Note: Answers from @marcsmith after I answered the questions. Sidenote: I think these questions were a little to complex. If I carry on with PQF (not sure if I will as coming up with Q's is hard), I'll probably look for stuff that can be answered in 1 or 2 paragraphs.
Quick
The proportion will remain the same. At first I considered it as Adrian did and that the number of boys would increase. However he has made the assumption that for the couple with the girl the next child would be a boy. The probability is still 50% in his scenario. That is the next child could be a girl or a boy. However even if there was a predilection for this pattern, i.e. when there is an odd number of couples, the odd couple?s next child will be a boy, the proportion would still remain the same. Let me explain this by increasing the data set. We'll start by simplifying and defining the rules:
- Every couple can have children,
- They stop when they have their first boy
- For every round of babies there is exactly a 50% ratio of boys to girls
- If there are an odd number of couples the odd couple will have a boy
- None of the children have their own children.
With these rules it follows that the number of boys, when all the couples in the current generation have had their children, will be the same as the number of couples. I'll use powers of two to keep the numbers simple.
For two couples the first iteration would produce 1 girl and 1 boy. That leaves one couple with a girl and therefore their next child will be a boy. So for 2 couples we have 2 boys and 1 girl.
For four couples the first iteration would produce 2 girls and 2 boys. The next iteration leaves 2 couples. We know from the previous example they produce 2 boys and 1 girl. So for 4 couple we get 4 boys and 3 girls.
For eight couples the first iteration would produce 4 girls and 4 boys. The next iteration leaves 4 couples. We know from the example the example with 4 couple that they produce 4 boys and 3 girls. So for 8 couple we get 8 boys and 7 girls.
Just by examining the examples we can see that for powers of 2 the number of boys is always one more than the number of girls. Let's tabulate this:
Couples Boys Girls Proportion Boys 2 2 1 66.67% 4 4 3 57.14% 8 8 7 53.33% 16 16 15 51.61% 32 32 31 50.79% 4096 4096 4095 50.01%
We can see that as the number of couples of increases the proportion tends towards 50% even though we always finish with boys.
Although we've produced rules that produce deterministic results it holds true if we use a random number generator to predict if a couple is having a boy or a girl with a 50% chance of it being either. Here are the results of a few consecutive runs for two couples from a quick simulation I wrote.
Couple 0: B, Couple 1: GB, Boys%: 66% Couple 0: GGB, Couple 1: B, Boys%: 50% Couple 0: B, Couple 1: GGB, Boys%: 50% Couple 0: B, Couple 1: B, Boys%: 100% Couple 0: B, Couple 1: GGGGB, Boys% 33% Couple 0: GB, Couple 1: GB, Boys%: 50%
As for the population itself. I suspect, and I could be wrong, that if everyone had to stop after they had one boy then for a 50% proportion the population would stagnate (that is after an initial growth so that each generation barring the last had all had their children). Anything other than 50% would eventually see a decline.
If there was no restriction on the number of boys then the population would most likely grow as long as the number of the smaller proportioned sex exceeded the number of couples in the previous generation but the proportion would still remain the same.
I could be wrong on the population stuff. I've only just considered it after Adrian's comment.
Cryptic
"I see it, but I don't believe it!" - Georg Cantor
The number of even natural numbers is exactly the same as the number of natural numbers. Every whole number can be mapped to an even number. No exceptions.
So 1=2, 2=4, 3=6, 4=8 ... n=2n
There is no case that exists where n exists and 2n does not. And for every natural number there is exactly one even integer that it maps to.
When there is exactly a one to one mapping between two sets it is called bijection. When one of the sets is the set of infinite natural numbers it is a countable set or a set that is denumerable.
It turns out a lot of things are denumerable:
- Even numbers
- Odd numbers
- Prime numbers
- Rational numbers (All integers and fractions)
That last one looks even more unlikely than the even numbers. Can there really be the same number of natural numbers as rational numbers. The set of rational numbers seems infinitely larger than the set of natural numbers. For every natural number as a numerator there is an infinite denominators. But looks can be deceiving. It turns out that every rational number can be mapped to a natural number in a one to one mapping.
Not all sets are denumerable. Irrational numbers, any real number that can't be expressed as a fraction, such as PI, are not countable.
This is because rational numbers settle into a pattern (1/2, 1/3, 1/4, 1/n) and can therefore be mapped whereas irrational numbers cannot, consider pi.
For more information and a much clearer explanation check out these links: