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Repair Windows errors before they cause bigger problemsFix Now →Scan for outdated or missing drivers - takes under a minuteDriver Scan →Clear out junk files and repair common Windows errorsFree Scan →Monte Carlo sampling estimates a probability or other quantity by repeatedly drawing outcomes from a probability model and averaging what happens. For example, simulate many groups of 100 fair-coin tosses, count how many groups have 45 or fewer heads, and divide that count by the number of groups simulated. The result approximates the probability of that event.
What Monte Carlo sampling means
Monte Carlo sampling is a way to estimate a target quantity using random samples. Instead of calculating a difficult probability, sum, or integral exactly, you draw outcomes from the relevant probability distribution, evaluate each outcome, and average the results.
For a quantity written as the expected value of a function of a random variable, the setup is:
Monte Carlo estimate = (1/n) Σᵢ₌₁ⁿ f(Xᵢ)
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- This guide is a perfect overview for the topics covered in introductory statistics courses.
Here, X₁ through Xₙ are independent draws from the distribution of X, and f(Xᵢ) is the value calculated for each draw. The estimate is the average of those values. This estimator is unbiased under the stated sampling setup, and it converges to the target as the sample count grows under the relevant conditions. The Deep Learning textbook’s Monte Carlo chapter derives the expectation estimator and discusses its convergence.
How random sampling estimates a probability
To estimate the probability of an event A, define f(X) as an indicator: it is 1 if the simulated outcome is in A and 0 otherwise. Averaging these zeroes and ones is the same as counting how often A occurred and dividing by the number of trials.
This turns probability estimation into a simple fraction:
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Estimated probability = successful simulated trials ÷ total simulated trials
SciPy’s statistics tutorial illustrates this event-fraction interpretation with a coin-toss example.
Worked example: 45 or fewer heads in 100 tosses
Suppose you want to estimate the probability of getting 45 or fewer heads in 100 tosses of a fair coin. The example parameters—head probability 0.5, 100 tosses, and a threshold of 45—define the question; they are not a reported empirical finding.
- Set the probability of heads to 0.5 and the number of tosses in each trial to 100.
- Simulate one group of 100 tosses and count its heads.
- Record whether that count is at most 45.
- Repeat the entire 100-toss trial many times, recording each trial that meets the condition.
- Divide the number of trials meeting the condition by the total number of trials.
There are two levels of repetition: the 100 tosses make one trial and produce one head count; repeating that full trial estimates the probability of the event. Increasing the number of tosses within one trial changes the event being studied, while increasing the number of trials gives more samples for estimating its probability. The example follows the setup in SciPy’s tutorial.
How many simulations do you need?
More independent samples generally make the estimate less noisy, but the improvement is gradual. For independent samples with finite variance, the variance of the sample mean is the variance of one sampled value divided by n. Its standard error therefore scales approximately as 1/√n.
The GNU Scientific Library documentation (GSL 2.8) describes this plain Monte Carlo error scaling: reducing error by a factor of 10 takes about 100 times as many sample points. Treat this as an order-of-magnitude planning rule, not a promise that each new run will be closer than the last. Random estimates fluctuate and need not improve monotonically.
No single sample count guarantees a chosen precision for every problem. A standard error or confidence interval requires assumptions and an appropriate calculation. For rare events, a simulation may observe few or no occurrences even when the event’s probability is not zero, so a simple estimate can be especially uncertain.
What the long-run law does—and does not—say
The law of large numbers explains why averages from suitable samples approach their expected value as the number of samples grows. It does not say that a short run must balance out, or that a particular result becomes due after a streak.
For independent fair-coin tosses, after five heads in a row the next toss still has a 50% chance of heads. The long-run statement concerns averages over an increasing number of tosses; it does not alter the probability of the next independent toss. Harvard’s probability text addresses this common misconception.
Assumptions and limits to keep in view
The basic explanation assumes independent draws from the intended probability distribution and finite variance for the values being averaged when using the standard error discussion. If the samples are dependent, biased, or drawn from the wrong distribution, simply increasing their number does not automatically make the answer valid. More samples can reduce random uncertainty without correcting a flawed model or systematic bias.
Not every method called a random simulation is Monte Carlo estimation: the defining feature here is using samples to estimate a target quantity. More advanced approaches—including Markov chain Monte Carlo and importance sampling—have different sampling assumptions and diagnostics; they should not be treated as interchangeable with direct independent draws.
Implementing a reproducible demonstration
For a coding example, NumPy recommends creating a random-number Generator with default_rng() and using it to draw from the needed distribution. NumPy describes these as pseudo-random numbers and provides seed controls. A seed makes a demonstration reproducible in its relevant software context; record the generator and seed if you need to reproduce a run. Do not assume identical random streams across software versions unless the specific version’s guarantee has been verified. See the NumPy random sampling documentation.
Quick Recap
Further reading
- Explorations in Monte Carlo Methods, described by Springer as covering probability, Monte Carlo experiments, and Python exercises. The publisher lists at least one year of calculus and a semester of matrix algebra as prerequisites, so it is not a required first step for every beginner.
- Introduction to Probability, an author-hosted course text described on the MIT course page as used in an introductory MIT course.
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