Gerolamo Cardano was celebrated throughout Europe as a physician to nobles of the Italian court and the Chair of Medicine at the University of Pavia. He was also something of an amateur mathematician.
In the early 1500s, as a young man, he liked to gamble - a pastime shared by many mathematicians before and since. But in 1564 or thereabouts, Cardano was perhaps the first to write down a systematic treatment of probability in his Book on Games of Chance (Liber de ludo aleae).
According to Leonard Mlodinow, in his book The Drunkard's Walk, much of Cardano's book is less than perfect, often reflecting the temperament of the author more than the technical aspects of the subject. Further, it only considers processes in which a single outcome is possible such as the toss of a die or the dealing of a playing card.
However, it did introduce one very important concept in Chapter 14 - the Law of Sample Spaces.
Translated into modern terms, it reads: "Suppose a random process has many equally likely outcomes, some favourable (that is, winning), some unfavourable (losing). Then the probability of obtaining a favourable outcome is equal to the proportion of outcomes that are favourable. The set of all possible outcomes is called the sample space."
Not the most easily interpreted language but in essence it says the possibility of winning or losing depends on the number of possible ways there are to win or to lose.
Consider tossing a single coin once.
The probability it will come down heads would appear to be equal to the probability it will come down tails. There are only two possible outcomes - H or T - and each represents exactly one half of all of the possible outcomes (H + T).
But consider tossing a coin twice or two coins simultaneously. How many possible outcomes can now occur?
A nave approach might be to say "three" - both heads, both tails, or one heads and one tails.
And on that response, many a successful gambler has made money.
In this case, there are actually four outcomes: HH, HT, TH, TT.
Each coin can still be heads or tails with a 50-50 chance. The four outcomes represent those possibilities.
If we look at this sample space, it is straightforward to see the likelihood of getting a combination of heads and tails is twice as likely as either both heads or both tails.
Put another way, there is a 50 per cent chance of the two coins coming up heads and tails and only a 25 per cent for the outcome of either both heads or both tails.
With three coins, the number of possible outcomes increases to eight: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. You are three times as likely to throw a combination of two heads and a tails as three heads.
This is what sample spaces tell us. The total number of possible outcomes can often be determined and from this, we can work out the likelihood of any given event.
For example, if a woman is carrying fraternal twins, what is the likelihood that one of the children will be a girl?
Perhaps not surprisingly, if we substitute girl for heads and boy for tails, we find there are four possible outcomes: (girl, girl), (girl, boy), (boy, girl) and (boy, boy). Of those outcomes, three will result in a girl so the probability is 75 per cent. What seems counter-intuitive is the probability one of the children will be a boy is also 75 per cent.
This is because the combination of boy and girl represents the result in 50 per cent of the possible outcomes.
Only in 25 per cent of the outcomes are two girls and 25 per cent two boys. But here is a follow up to this question: given that one child in a pair of fraternal twins is a girl, what are the chances the other child will also be a girl?
The gut response would be to say 50-50. After all, the other child is either a boy or a girl.
But consider our sample space.
Nothing has changed except we have eliminated the possibility of a (boy, boy) outcome. The other three possible outcomes still remain and two of them involve a boy and girl combination.
Hence, the likelihood of a (girl, girl) outcome is only one in three or 33 per cent.
If you are betting on the answer, a boy and girl outcome gives you the best odds.
Notice though that the question didn't specify birth order.
If I had worded the question by saying "If the first child born of fraternal twins is a girl, what are the chances the other child will also be a girl?" then the odds switch back to 50-50.
Why? Because we have also eliminated (boy, girl) from our sample space.
Understanding sample space is critical to understanding probability.
Unfortunately, it is not always easy to do in real world problems.
