Probability Theory Concepts

Two events are independent if knowing that one event happened does not change the probability of the other event.

"Independence means “no update.” If learning B happened leaves the chance of A exactly the same, then the events are independent."

Why it matters: Independence is the dividing line between two completely different probability tools. If events are independent, you can multiply their probabilities directly — that's how every coin/die/spinner/dart problem becomes simple. If they are NOT independent, you must use conditional probability and the multiplication rule with $P(A \mid B)$, which is much harder. Getting this distinction wrong is the single most common source of error on probability tests.

The addition rule finds the probability that at least one of two events occurs. It adds the probabilities of the two events and then subtracts any overlap so the shared outcomes are not counted twice.

"If you want “A or B,” start by adding A and B. Then fix the double-counting by removing the part that belongs to both events."

Why it matters: Addition Rule helps students reason about uncertainty without guessing. It connects outcomes, sample spaces, and event rules so students can decide whether to add, multiply, condition, simulate, or compare long-run behavior.

The multiplication rule finds the probability that two events both occur. It multiplies the probability of the first event by the conditional probability of the second event given that the first has happened.

"For an “and” problem, move through the events in sequence. Take the chance of the first step, then update for the second step based on what is already known."

Why it matters: Multiplication Rule helps students reason about uncertainty without guessing. It connects outcomes, sample spaces, and event rules so students can decide whether to add, multiply, condition, simulate, or compare long-run behavior.

The Law of Large Numbers states that as the number of independent, identically distributed trials increases, the sample average converges to the theoretical expected value (population mean). In other words, larger samples produce more reliable estimates of the true probability or average.

"Flip a coin 10 times: maybe 7 heads (70%). Flip 100 times: closer to 50%. Flip 10,000 times: very close to 50%. More trials = more reliable averages. Short-run luck evens out."

Why it matters: Law of Large Numbers helps students reason about uncertainty without guessing. It connects outcomes, sample spaces, and event rules so students can decide whether to add, multiply, condition, simulate, or compare long-run behavior.

The expected value of a random variable is the long-run average outcome of a random process, calculated as the weighted sum of each possible outcome times its probability. It represents what you would earn or lose on average per trial if the process were repeated infinitely many times.

"If you played a game forever, expected value is your average result per play. Positive EV = profitable long-term. Negative EV = you'll lose over time. It's the mathematical way to evaluate risky decisions."

Why it matters: Expected Value helps students reason about uncertainty without guessing. It connects outcomes, sample spaces, and event rules so students can decide whether to add, multiply, condition, simulate, or compare long-run behavior.