Probability can be approached from two directions. Theoretical probability is what you reason out in advance: if a process has equally likely outcomes, you count the outcomes favorable to an event and divide by the total number in the sample space. Experimental (or empirical) probability is what you measure: you run or observe trials and divide the number of times the event actually occurred by the number of trials performed. Both produce a number between 0 and 1, but they are obtained in fundamentally different ways and they carry different assumptions.
This distinction matters in engineering because many real systems do not come with a tidy list of equally likely outcomes. You rarely know the true failure probability of a component from first principles, so you estimate it from test data or field observations. Conversely, when a model is available, theoretical probability lets you predict behavior before building anything. The bridge between the two is the Law of Large Numbers, which says that as the number of independent trials grows, the relative frequency of an event tends to settle near its true probability. That is why a short run of 10 coin flips can look lopsided while 10,000 flips look nearly even.
As you watch, keep three questions in mind. First, how is each probability computed — by counting outcomes in a sample space, or by counting occurrences in data? Second, what does the Law of Large Numbers actually promise, and what does it not promise (it does not say that a run of bad luck must be "balanced out")? Third, when experimental results disagree with theoretical predictions, what are the plausible explanations: too few trials, a biased or unfair process, or a flawed model assumption? Being able to argue which explanation applies is a core engineering skill.
A short, direct introduction that defines experimental and theoretical probability side by side and walks through how to compute each. Based on its title and description, it focuses on the contrast between the two approaches and shows worked examples of calculating probabilities both ways. Expect simple, familiar settings such as dice, coins, or spinners used to make the comparison concrete.
At roughly four minutes it is the most efficient entry point for the first three subtopics — defining each type of probability and naming the key differences — and gives you shared vocabulary before the longer, more mathematical treatments.
Steve Brunton presents the Law of Large Numbers as a formal result in probability, framing it as a stepping stone toward broader results such as the Central Limit Theorem. The video is likely to cover what convergence of a sample average means, why repeated independent trials drive the observed frequency toward the true probability, and the conditions under which the result holds. Expect a mix of intuition and mathematical statement rather than arithmetic drills.
Brunton is a mechanical engineering professor known for clear, engineering-oriented treatments of applied mathematics, making this a strong fit for the Law of Large Numbers subtopic at the level expected in an engineering probability course.
A lecture segment from a data science course that explains the Law of Large Numbers in the context of working with data and simulations. It is likely to show how an empirical estimate behaves as the number of observations increases, illustrating convergence through computation rather than proof. The framing emphasizes practical use of sampled data.
It complements the more theoretical treatment by showing the data-driven side of the same idea, supporting the subtopics on practical applications and on comparing experimental results against theoretical values.
Together these videos target MO3 most directly: the first clarifies the definitions and derivation methods of theoretical versus experimental probability, while the two Law of Large Numbers videos explain the principle that links an observed relative frequency to an underlying theoretical probability. Computing theoretical probability requires enumerating outcomes, so the examples reinforce the sample-space thinking in MO1 and the basic rule application in MO2 and MO5. The data-oriented treatment of convergence also supports MO4 by showing how a probability estimate drawn from trials should be stated and interpreted — including how much confidence a given number of trials actually warrants in a real decision-making setting.
These videos were chosen from YouTube on October 2, 2026. Each has captions; the player turns them on. Videos on YouTube can be changed or removed — if one no longer plays, tell your instructor. If a player does not appear, use its Watch on YouTube link.