Probability measures how likely an event is to occur. It is always a number between 0 and 1 (or between 0% and 100%). A probability of 0 means the event is impossible (like rolling a 7 on a standard six-sided die). A probability of 1 means the event is certain (like the sun rising tomorrow). A probability of 0.5 (50%) means the event is equally likely to happen or not happen (like flipping a coin and getting heads). The formula for probability is: P(event) = (number of favorable outcomes) ÷ (total number of possible outcomes). If you roll a fair six-sided die (numbers 1 through 6), what is the probability of rolling an even number (2, 4, or 6)?

Complementary events are two outcomes that cover all possibilities – one happens or the other happens, but not both. The sum of the probabilities of complementary events always equals 1 (or 100%). For example, the complement of "rolling a 6" is "not rolling a 6" (rolling 1,2,3,4, or 5). If the probability of rain tomorrow is 0.3 (30%), what is the probability of NO rain tomorrow (the complement)?

Theoretical probability is what SHOULD happen based on mathematics. Experimental probability is what ACTUALLY happens when you perform an experiment. The Law of Large Numbers says that as you repeat an experiment many times, the experimental probability gets closer to the theoretical probability. You flip a fair coin 100 times and get heads 45 times and tails 55 times. What is the experimental probability of getting heads? What is the theoretical probability?

Independent events do not affect each other. The outcome of one event has no influence on the outcome of another. For independent events, the probability of BOTH happening is found by multiplying their individual probabilities: P(A and B) = P(A) × P(B). For example, rolling a die and flipping a coin are independent. What is the probability of rolling a 6 on a fair die AND flipping heads on a fair coin?

Mutually exclusive events cannot happen at the same time. For example, drawing a single card from a deck cannot be both a King and a Queen at the same time. For mutually exclusive events, the probability of EITHER event happening is found by adding their individual probabilities: P(A or B) = P(A) + P(B). You draw one card from a standard 52-card deck. What is the probability of drawing either a King OR a Queen? (There are 4 Kings and 4 Queens.)

Dependent events are affected by previous outcomes. When you draw without replacement, the probabilities change because the total number of outcomes decreases. A bag contains 5 red marbles and 3 blue marbles (8 total). You draw one marble, do NOT put it back, then draw a second marble. What is the probability of drawing a red marble first AND a blue marble second?

Expected value (or expectation) is the long-run average outcome of a random event. To calculate expected value, multiply each possible outcome by its probability, then add all these products together. In a simple dice game, you win $10 if you roll a 6, and you lose $2 if you roll any other number (1,2,3,4,5). What is the expected value of playing this game once? (This tells you how much you would expect to win or lose on average per game over many plays.)

Calculating "at least one" probabilities is often easier using the complement rule. P(at least one) = 1 - P(none). For example, if you flip a coin 3 times, what is the probability of getting at least one head? Instead of listing all possibilities with at least one head (7 outcomes), find P(no heads) = P(all tails) = (1/2)³ = 1/8, then subtract from 1. A family has 4 children. Assuming boys and girls are equally likely (each 1/2) and independent, what is the probability that the family has at least one boy?

Odds are another way to express likelihood, but odds are NOT the same as probability. Odds compare the number of favorable outcomes to the number of unfavorable outcomes. "Odds in favor" = favorable : unfavorable. "Odds against" = unfavorable : favorable. If the probability of an event is 1/4 (25%), that means for every 1 favorable outcome, there are 3 unfavorable outcomes. So odds in favor are 1:3 (read as "1 to 3"). If a horse has odds of 5:1 to win a race (odds in favor), what is the probability that the horse wins?

Some probabilities are surprising and counterintuitive! The Birthday Paradox asks: How many people do you need in a room for there to be a greater than 50% chance that at least two share the same birthday (month and day, ignoring year)? Most people guess 183 (half of 365), but the answer is only 23! This is because we are comparing all pairs of people, not matching one person's birthday. With 23 people, there are 253 pairs, making the probability much higher than expected. If you have 23 people in a room, what is the approximate probability that at least two share a birthday? (This is a famous result – just remember it is over 50%!)