13.7 Probability

Probability Fundamentals

Probability gives you a way to measure how likely something is to happen, using a number between 0 and 1. These tools show up constantly in statistics, science, and finance, so getting comfortable with them now pays off.

Basic Probability Model

Probability measures the likelihood of an event on a scale from 0 (impossible) to 1 (certain). To calculate it, you need two things: a sample space and an event.

• Sample space (S) is the set of all possible outcomes. For rolling a die, $S = \{1, 2, 3, 4, 5, 6\}$.
• Event (E) is a subset of the sample space you care about. "Rolling an even number" means $E = \{2, 4, 6\}$.

The probability of an event is:

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

For example, the probability of drawing a heart from a standard 52-card deck is $\frac{13}{52} = \frac{1}{4}$, since there are 13 hearts out of 52 total cards.

Equal Likelihood Probabilities

When every outcome in the sample space is equally likely, you can apply the formula above directly:

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

• Rolling a 3 on a fair six-sided die: $P(3) = \frac{1}{6}$ (one favorable outcome, six total)
• Flipping heads on a fair coin: $P(\text{heads}) = \frac{1}{2}$ (one favorable outcome, two total)

The key word here is fair. This formula only works when no outcome is more likely than another.

Probability Rules and Techniques

Union Rules for Combined Events

The union of events A and B, written $A \cup B$, is the event that at least one of them occurs. The formula for its probability is:

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

You subtract $P(A \cap B)$ (the probability both happen) because otherwise you'd count that overlap twice.

Mutually exclusive events can't happen at the same time, so $P(A \cap B) = 0$. The formula simplifies to:

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

For example, drawing a heart or a spade from a deck: these suits don't overlap, so $P(\text{heart or spade}) = \frac{13}{52} + \frac{13}{52} = \frac{1}{2}$.

Independent events are different from mutually exclusive events. Two events are independent when the occurrence of one doesn't change the probability of the other. For instance, flipping a coin and rolling a die are independent because the coin result has no effect on the die result.

Complement Rule

The complement of event A, written $A'$ or $A^c$, is everything in the sample space that isn't A. Its probability is:

$P(A') = 1 - P(A)$

This is especially useful when it's easier to calculate the probability of something not happening. A couple of examples:

• Probability of not drawing a red card: $1 - \frac{1}{2} = \frac{1}{2}$
• Probability of not rolling a 6: $1 - \frac{1}{6} = \frac{5}{6}$

Counting Techniques for Complex Probabilities

When problems get more involved, you need systematic ways to count outcomes. There are three main tools.

Fundamental Counting Principle: If one event can happen in $m$ ways and an independent second event can happen in $n$ ways, the two events together can happen in $m \times n$ ways. A menu with 3 appetizers and 4 entrees gives $3 \times 4 = 12$ possible meals. This extends to any number of events.

Permutations count arrangements where order matters.

• The number of ways to arrange all $n$ distinct objects is $n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$. Arranging 5 books on a shelf: $5! = 120$.
• The number of ways to arrange $r$ objects chosen from $n$ is: $P(n, r) = \frac{n!}{(n-r)!}$. Selecting 3 people from 10 to stand in a line (where position matters): $P(10, 3) = \frac{10!}{7!} = 720$.

Combinations count selections where order does not matter.

$C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$

Choosing 3 people from 10 for a committee (where position doesn't matter): $C(10, 3) = \frac{10!}{3! \cdot 7!} = 120$. Notice this is smaller than the permutation answer because we're no longer distinguishing the order.

Using counting in probability: Once you can count favorable and total outcomes, plug them into the probability formula. For example, the probability of drawing 2 aces from a deck in 2 draws without replacement:

$P(\text{2 aces}) = \frac{C(4, 2)}{C(52, 2)} = \frac{6}{1326} = \frac{1}{221}$

The numerator counts ways to choose 2 aces from the 4 available. The denominator counts all ways to choose any 2 cards from 52.