The mathematics of uncertainty - from a simple coin toss to random sampling, expected value and the foundations of statistical modelling.
Probability gives us a mathematical way of describing how likely an event is to occur. It ranges from 0, meaning impossible, to 1, meaning certain.
For a fair six-sided die, each number has an equal probability of appearing.
Probability becomes especially interesting when we repeat an experiment many times. Individual outcomes may look unpredictable, but patterns begin to emerge across large numbers of observations.
A fair coin has two equally likely outcomes: heads or tails. The theoretical probability of heads is therefore 50%.
If we repeatedly flip a fair coin, the proportion of heads tends to move towards 50%.
The individual flips remain random. What changes is our confidence in the long-run proportion.
Expected value is one of the most useful ideas in probability. It represents the long-run average outcome of a random process.
Consider a fair six-sided die. Each outcome has probability 1/6, so the expected value is:
Suppose we have a population of thousands of observations. Instead of analysing every observation, we can take a random sample.
The sample will not perfectly reproduce the population, but probability tells us something about how reliable that sample is likely to be.
The complete set of observations we are interested in. For example, every transaction, borrower or investment return in a dataset.
A subset selected from the population. Repeated random samples allow us to understand the distribution of possible outcomes.
Probability is the foundation beneath much of modern statistics and quantitative modelling.
Many naturally occurring measurements can be modelled using the bell-shaped normal distribution.
Standard deviation measures how widely observations are distributed around their mean.
Once we can generate random observations from a probability distribution, we can simulate thousands or millions of possible futures.
This is the basic idea behind Monte Carlo simulation. Instead of asking:
we ask:
That simple shift in perspective connects probability to finance, risk management, forecasting, portfolio modelling and many of the quantitative tools used elsewhere on KerryTech.
Probability does not allow us to predict every individual event. A fair coin can still produce ten heads in a row.
Its power comes from describing the structure of uncertainty across many possible outcomes.
A single event can be unpredictable. A large collection of random events can be remarkably predictable.
That idea sits at the heart of statistics, financial modelling, insurance, economics, science and machine learning.