3.1 Terminology

Probability Terminology and Concepts

Probability gives us a framework for measuring how likely something is to happen. Before you can calculate anything, you need to know the core vocabulary: experiments, outcomes, sample spaces, and events. This section covers those building blocks.

Key Probability Terminology

An experiment is any process that produces well-defined outcomes. Rolling a die, flipping a coin, or drawing a card from a deck are all experiments. The key requirement is that you can clearly list what results are possible.

An outcome is a single result of an experiment. Rolling a 3, flipping tails, or drawing the Ace of Spades are each one outcome.

The sample space is the set of all possible outcomes, denoted by $S$. For a single die roll: $S = \{1, 2, 3, 4, 5, 6\}$. For a coin flip: $S = \{H, T\}$. Every probability question starts with knowing the sample space.

An event is any subset of the sample space. It can be a single outcome or a collection of outcomes. "Rolling an even number" is the event $\{2, 4, 6\}$, which contains three of the six outcomes in the sample space.

• Random variable: a function that assigns a numerical value to each outcome in the sample space. For example, if you flip two coins, you could define a random variable $X$ that counts the number of heads, so $X$ could equal 0, 1, or 2.

Probability Calculation for Equally Likely Events

When every outcome in the sample space has the same chance of occurring (like with a fair die or a fair coin), you can calculate probability with a straightforward formula:

$P(A) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$

The probability of rolling a 3 on a fair die is $\frac{1}{6}$ because there's 1 favorable outcome out of 6 total. The probability of drawing a red card from a standard deck is $\frac{26}{52} = \frac{1}{2}$ because 26 of the 52 cards are red.

Two important rules follow from this:

• The probability of the entire sample space is always 1: $P(S) = 1$. Something has to happen.
• The complement of an event $A$, written $A^c$, represents everything in the sample space that is not in $A$. Its probability is $P(A^c) = 1 - P(A)$. So if the probability of rolling a 3 is $\frac{1}{6}$, the probability of not rolling a 3 is $1 - \frac{1}{6} = \frac{5}{6}$.

"And" vs. "Or" Events

These two words have precise meanings in probability, and mixing them up is one of the most common mistakes.

"AND" (Intersection): $A \cap B$ means both events occur. Rolling a 3 and flipping heads requires both things to happen.

• If events are independent (one doesn't affect the other): $P(A \cap B) = P(A) \times P(B)$
• Example: $P(\text{rolling a 3 AND flipping heads}) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$

"OR" (Union): $A \cup B$ means at least one of the events occurs. You use the Addition Rule:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

You subtract $P(A \cap B)$ to avoid double-counting outcomes that belong to both events.

Mutually exclusive events cannot happen at the same time. Rolling a 3 and rolling a 5 on a single die roll are mutually exclusive because you can't get both at once. When events are mutually exclusive, $P(A \cap B) = 0$, so the formula simplifies to:

$P(A \cup B) = P(A) + P(B)$

Additional Probability Concepts

Conditional probability is the probability of an event occurring given that another event has already happened. It's written $P(A|B)$, read as "the probability of A given B." For example, the probability of drawing a king from a deck changes if you already know the card is a face card.

Independence means two events don't influence each other. If knowing that $B$ happened doesn't change the probability of $A$, then $A$ and $B$ are independent. Coin flips are a classic example: the result of the first flip has no effect on the second.

The Law of Large Numbers says that as you repeat an experiment more and more times, the experimental probability (what you actually observe) gets closer and closer to the theoretical probability. Flip a coin 10 times and you might get 70% heads. Flip it 10,000 times and you'll be much closer to 50%.