In the 16th century,
an Italian mathematician named Cardano
had a well-known gambling addiction.
He wrote letters in which he bragged
about his ability to beat his friends at dice games.
His trick was to place his bets
using his ideas from mathematics –
instead of hunches or luck.
He came up with the method of calculating
the exact probability of random events,
such as, say, rolling 'snake eyes' – (a pair of 1's).
It is based on a powerful property Cardano noticed.
Every outcome – no matter how many dice you roll –
is equally likely.
This allowed him to calculate the probability
by developing what's now called a 'probability space.'
First, he counts all possible outcomes –
known as the 'sample space.'
For a single dice, these are the 6 possible faces.
Then, he defines the event in question –
such as rolling a one –
which can occur in one way.
The probability is then found
by dividing the event by all possible events
which in this case works out to 1/6.
Realize the cold, calculating logic here.
There is no such things as a 'lucky number' –
no 'divine intervention.'
The probability of rolling any number is exactly 1/6.
Now the same logic applies when we roll multiple dice.
Imagine he needed to know the probability of rolling a pair.
First, he'd count the size of the sample space.
With two dice rolls,
there are 36 possible outcomes: 6 x 6.
Then, he'd count the number of ways this event can occur.
There are six different pairs.
So, 6 divided by 36 is the probability of rolling a pair –
also 1/6.
This simple yet powerful idea allowed Cardano
to bet according to the true probability –
while his opponents placed their bets
based on hunches and lucky numbers.
Remember, this works with multiple rolls.
[So,] Imagine we needed to know
the exact probability of rolling three 1's.
Simple!
First, we'd figure out the size of the sample space.
For three dice, this is 6 x 6 x 6, – or 216.
And there is only one way to roll three 1's,
so, the probability is 1 divided by 216 (or 1/216).
This was the trick:
It was not based on magic, but mathematics.
Remember, to calculate the probability of a random event –
such as a dice roll –
you divide the number of ways that event can occur
by all possible outcomes.
Learning!
This is fascinating stuff.