Probability Basics

Sample spaces and events

The sample space $S$ is the set of all possible outcomes. An event is a subset of $S$.

If all outcomes are equally likely:

$P(\text{event}) = \frac{\text{number of outcomes in event}}{\text{total outcomes in } S}$

Example. Roll a fair 6-sided die. $S = \{1, 2, 3, 4, 5, 6\}$. The event "even" is $\{2, 4, 6\}$, so $P(\text{even}) = 3/6 = 1/2$.

Probability axioms (the rules of the game)

• $0 \leq P(A) \leq 1$
• $P(S) = 1$
• If $A$ and $B$ are mutually exclusive: $P(A \text{ or } B) = P(A) + P(B)$

Complements

$P(\text{not } A) = 1 - P(A)$

Often easier to compute the complement than the event itself.

Independent events

Events $A$ and $B$ are independent if knowing one happened doesn't change the probability of the other. For independent events:

$P(A \text{ and } B) = P(A) \cdot P(B)$

Example. Flip a fair coin twice. $P(\text{HH}) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$.

Conditional probability

The probability of $A$ given $B$ already happened:

$P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}$

Example. Draw one card from a standard deck. What's the probability it's a king given it's a face card? There are 12 face cards, 4 of which are kings: $P(\text{King} \mid \text{Face}) = 4/12 = 1/3$.

Expected value

For a random variable $X$ taking values $x_1, x_2, \ldots$ with probabilities $p_1, p_2, \ldots$:

$E[X] = \sum_i x_i \cdot p_i$

Example — a $1 bet on a die roll. You pay $1 to play. If you roll a 6, you win $5 (net $+4$). Otherwise you lose your $1 (net $-1$).

$E[\text{profit}] = (1/6)(4) + (5/6)(-1) = 4/6 - 5/6 = -1/6$

You lose about 17 cents per play on average — don't take this bet.

Key takeaways

• Probability for equally likely outcomes = favorable / total.
• $P(\text{not } A) = 1 - P(A)$ — the complement trick saves time.
• Independent: multiply. Mutually exclusive (and not both can happen): add.
• Expected value tells you the average outcome over many trials.