Chapter 14: Probability

Probability quantifies uncertainty. When you toss a coin, you don't know whether it will land heads or tails , but you do know the chance of heads is $1/2$. When you roll a die, the chance of getting a $6$ is $1/6$. When you draw a card from a well-shuffled deck, the chance of drawing the ace of spades is $1/52$. Probability assigns a number between $0$ and $1$ to every possible event, capturing how likely it is to occur.

This chapter focuses on the theoretical (or classical) approach: when the outcomes of an experiment are equally likely, the probability of an event is the ratio of favourable outcomes to total outcomes. This is the standard board-exam framework. You'll learn to identify sample spaces (the set of all possible outcomes), count favourable outcomes systematically, and compute probabilities for coins, dice, cards, and balls in bags.

Probability has enormous real-world relevance: predicting weather, designing insurance policies, evaluating medical tests, gambling, sports analytics, machine learning. The reasoning style , count carefully, then divide , is the foundation of all probabilistic thinking. The more carefully you list outcomes, the fewer mistakes you make.

Board exam questions are usually short and direct. You'll be given a setup ("a card is drawn from a well-shuffled deck"), asked the probability of an event ("getting a red king"), and asked to express it as a fraction in lowest terms. The technique is the same throughout: identify favourable outcomes, divide by total outcomes.

What's inside

• Sample space , list of all possible outcomes.
• Theoretical probability , formula and interpretation.
• Events , simple and compound.
• Complementary events , $P(\bar A) = 1 - P(A)$.
• Cards, coins, dice, and balls , standard examples.

Key results / Formula card

• Probability formula: $P(E) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$, when all outcomes are equally likely.
• $0 \leq P(E) \leq 1$ for any event $E$.
• $P(\text{certain event}) = 1$; $P(\text{impossible event}) = 0$.
• Complement: $P(\bar E) = 1 - P(E)$, where $\bar E$ is "$E$ does not occur".
• Standard deck of cards: $52$ cards = $4$ suits (hearts, diamonds, clubs, spades) $\times 13$ ranks (Ace, $2, 3, \ldots, 10$, Jack, Queen, King). Red: hearts and diamonds. Black: clubs and spades. Face cards: Jack, Queen, King = $12$ cards.
• Standard die: $6$ faces labelled $1$ through $6$.
• Coin: two outcomes, heads or tails.