3.3 Two Basic Rules of Probability

Probability Rules

Multiplication Rule for Probabilities

The multiplication rule answers the question: what's the probability that two (or more) events both happen? Anytime you see "and" in a probability problem, this is likely the rule you need.

For independent events (where one event doesn't affect the other):

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

Two events are independent when the outcome of one has no effect on the probability of the other. Rolling a die and flipping a coin is a classic example. Getting a 4 on the die doesn't change your chances of getting heads.

Say you want the probability of rolling a 3 and flipping heads. That's $P(3) \times P(H) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$.

For dependent events (where one event changes the probability of the other):

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

The notation $P(B|A)$ means "the probability of B given that A already happened." This is called conditional probability.

A common example: drawing two cards from a standard deck without replacement. The probability of drawing an ace first is $\frac{4}{52}$. If that happened, the probability of drawing a second ace is now $\frac{3}{51}$, because one ace and one card are gone. So $P(\text{two aces}) = \frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} \approx 0.0045$.

Addition Rule in Probability

The addition rule answers a different question: what's the probability that at least one of these events occurs? Look for the word "or" in the problem.

The general formula is:

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

You subtract $P(A \text{ and } B)$ to avoid double-counting outcomes that fall in both events.

For mutually exclusive events (events that can't happen at the same time), the formula simplifies:

$P(A \text{ or } B) = P(A) + P(B)$

There's nothing to subtract because $P(A \text{ and } B) = 0$. For example, a single die roll can't be both even and odd, so $P(\text{even or odd}) = \frac{3}{6} + \frac{3}{6} = 1$.

For non-mutually exclusive events (events that can happen at the same time), you must use the full formula. Drawing a heart or a face card from a deck is a good example. Some cards are both hearts and face cards (the jack, queen, and king of hearts), so you'd double-count those three if you just added.

$P(\text{heart or face card}) = \frac{13}{52} + \frac{12}{52} - \frac{3}{52} = \frac{22}{52} \approx 0.423$

Venn diagrams are a great way to visualize this overlap and make sure you're not double-counting.

Multiplication vs. Addition Rules

Choosing the right rule comes down to reading the problem carefully.

• Multiplication rule ("and"): You want the probability of multiple events all happening.
  • Keywords: "and," "both," "given that," "if"
  • Then decide: are the events independent or dependent?
• Addition rule ("or"): You want the probability of at least one event happening.
  • Keywords: "or," "either," "at least one"
  • Then decide: are the events mutually exclusive or not?

A quick decision process:

1. Read the problem and identify whether it's asking about events happening together or at least one happening.
2. Pick the multiplication rule (together) or addition rule (at least one).
3. Determine the relationship between the events (independent/dependent, or mutually exclusive/not).
4. Apply the correct version of the formula.

Fundamental Concepts in Probability

A few foundational ideas tie everything together:

• Sample space: the set of all possible outcomes. For a coin flip, it's {heads, tails}. For a die roll, it's {1, 2, 3, 4, 5, 6}. Knowing your sample space helps you count outcomes correctly.
• Complementary events: two events that cover every outcome in the sample space with no overlap. "Rolling a 6" and "not rolling a 6" are complements. Their probabilities always add to 1:

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

This is useful when it's easier to calculate the probability of something not happening. Just find $P(A')$ and subtract from 1.

• Law of total probability: a way to find the probability of an event by breaking it into separate cases that cover all possibilities, then adding up the probabilities of each case. You'll see this applied more in later topics, but the core idea is that you can split a hard problem into simpler pieces.