Randomness, Random Numbers and Monte Carlo Simulation

Space missions are designed with randomness on purpose. Simulating thousands of slightly different flights is how planners see the range of what could happen.

What "random" means

An event is random when we cannot predict its individual outcome, even though we may know the pattern of outcomes over many trials very well. A fair die is the standard example: no one can say what the next roll will be, but over many rolls each face appears about a sixth of the time.

Two rolls of a fair die are independent: the result of one tells you nothing about the next. This simple idea is easy to state and surprisingly hard to feel, as we will see below.

How computers make random numbers

Random number generation comes in two broad kinds.

• A pseudorandom number generator is an algorithm. Given a starting value, called a seed, it produces a sequence of numbers that looks random and passes statistical tests, but is entirely determined by the seed. The same seed gives the same sequence, which is useful for repeating a simulation exactly.
• A hardware random number generator draws randomness from a physical process, such as electrical noise, rather than from an algorithm.

For simulations, pseudorandom numbers are usually what you want, because results can be reproduced and checked.

The Monte Carlo method

Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The approach was conceptualised by the mathematician Stanisław Ulam, and the underlying idea is to use randomness to solve problems that may themselves be deterministic.

In mission design, it works like this. A real launch will not go exactly to plan: thrust varies a little, winds vary, the spacecraft's mass is known only within a tolerance. Engineers describe each uncertain input as a range with a probability distribution, then run the flight simulation thousands of times, each time drawing new random values. The spread of results shows how likely the mission is to hit its target corridor and how much margin it has.

Draw x, y uniformly between 0 and 1, many times.
Count the fraction with x² + y² ≤ 1.
That fraction approaches π / 4.

The gambler's fallacy

The most common mistake about randomness has a name. The gambler's fallacy is the belief that an independent outcome which has happened less often than expected recently is more likely to happen next, as if the dice were due. For a fair die, the chance of a four on the next roll is one in six, whatever happened on the last ten rolls. The same fallacy turns up in engineering when people assume that a long run of successful launches means a failure is "due", or, in reverse, that it cannot happen now.

Randomness and decisions

Knowing that outcomes are uncertain is only half the job. The other half is deciding what to do about it, which is where expected value comes in.

REV 2026-11-09 / Tobias Wren