Basic Probability

Probability measures how likely an event is to occur, on a scale from 0 (impossible) to 1 (certain). It connects the counting techniques from counting principles and permutations and combinations to real-world decision-making. In this lesson you will learn the basic probability formula, complementary events, the rules for combining events (independent and mutually exclusive), and a brief introduction to expected value.

The Basic Probability Formula

For an experiment with equally likely outcomes:

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

Probability is always between 0 and 1 (inclusive). A probability of 0.5 (or 50 percent) means the event is equally likely to happen or not.

Example 1: Rolling a Die

What is the probability of rolling a 4 on a standard six-sided die?

$P(4) = \frac{1}{6} \approx 0.167$

Example 2: Drawing a Card

What is the probability of drawing a heart from a standard 52-card deck?

$P(\text{heart}) = \frac{13}{52} = \frac{1}{4} = 0.25$

Example 3: Using Combinations

A jar contains 6 red marbles and 4 blue marbles. If you draw 3 marbles at random, what is the probability that all 3 are red?

Total ways to draw 3 from 10: $C(10, 3) = 120$

Ways to draw 3 red from 6: $C(6, 3) = 20$

$P(\text{all red}) = \frac{20}{120} = \frac{1}{6} \approx 0.167$

Complementary Events

The complement of event $E$ (written $E'$ or $\bar{E}$) is “event $E$ does NOT happen.” Since the event either happens or it does not:

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

This is extremely useful when calculating “at least one” probabilities — it is often easier to compute the probability of none and subtract.

Example 4: At Least One Head

A coin is flipped 4 times. What is the probability of getting at least one head?

$P(\text{no heads}) = P(\text{all tails}) = \left(\frac{1}{2}\right)^4 = \frac{1}{16}$

$P(\text{at least one head}) = 1 - \frac{1}{16} = \frac{15}{16} = 0.9375$

Example 5: Defective Products

A batch of 50 light bulbs has 3 defective ones. If you randomly select 5, what is the probability that at least one is defective?

$P(\text{none defective}) = \frac{C(47, 5)}{C(50, 5)} = \frac{1{,}533{,}939}{2{,}118{,}760} \approx 0.7240$

$P(\text{at least one defective}) = 1 - 0.7240 = 0.2760$

Independent Events

Two events are independent if the occurrence of one does not affect the probability of the other. For independent events:

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

Multiply probabilities for “and” with independent events.

Example 6: Two Dice

What is the probability of rolling a 5 on the first die AND a 6 on the second?

$P(5 \text{ and } 6) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$

Example 7: Free Throws

A basketball player makes 80 percent of free throws. What is the probability of making 3 in a row?

$P(\text{3 in a row}) = 0.80 \times 0.80 \times 0.80 = 0.512$

There is about a 51 percent chance.

Example 8: Independent with Complement

The probability that a server crashes on any given day is 0.02. If two servers operate independently, what is the probability that at least one server is running?

$P(\text{both crash}) = 0.02 \times 0.02 = 0.0004$

$P(\text{at least one running}) = 1 - 0.0004 = 0.9996$

Mutually Exclusive Events

Two events are mutually exclusive (or disjoint) if they cannot happen at the same time. For mutually exclusive events:

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

Add probabilities for “or” with mutually exclusive events.

Example 9: Rolling a Die

What is the probability of rolling a 2 or a 5?

$P(2 \text{ or } 5) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$

Non-Mutually-Exclusive Events: The General Addition Rule

When events can overlap, subtract the double-counted intersection:

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

Example 10: Cards

What is the probability of drawing a king or a heart from a 52-card deck?

$P(\text{king}) = \frac{4}{52}$, $P(\text{heart}) = \frac{13}{52}$, $P(\text{king of hearts}) = \frac{1}{52}$

$P(\text{king or heart}) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13} \approx 0.308$

Using Combinations in Probability

Many probability problems reduce to “favorable combinations / total combinations.”

Example 11: Poker Hand

What is the probability of being dealt exactly 2 aces in a 5-card hand?

Choose 2 aces from 4: $C(4, 2) = 6$

Choose 3 non-aces from 48: $C(48, 3) = 17{,}296$

Total 5-card hands: $C(52, 5) = 2{,}598{,}960$

$P(\text{exactly 2 aces}) = \frac{6 \times 17{,}296}{2{,}598{,}960} = \frac{103{,}776}{2{,}598{,}960} \approx 0.0399$

About a 4 percent chance.

Expected Value (Brief Introduction)

The expected value is the long-run average outcome of a random process:

$E(X) = \sum_{i} x_i \cdot P(x_i)$

where $x_i$ are the possible values and $P(x_i)$ is the probability of each.

Example 12: Game Show

A game show has prizes: win $100 with probability 0.2, win $50 with probability 0.3, win $0 with probability 0.5. What is the expected winnings?

$E(X) = 100(0.2) + 50(0.3) + 0(0.5) = 20 + 15 + 0 = 35$

On average, a contestant wins $35 per game.

Example 13: Fair Price for a Raffle

A raffle sells 500 tickets. The prize is $1,000. What is the expected value of a single ticket?

$E(X) = 1000 \times \frac{1}{500} + 0 \times \frac{499}{500} = 2$

The expected value is $2, so a ticket priced above $2 is not “fair” in a purely mathematical sense.

Real-World Application: Nursing — Medication Error Rates

A hospital administers medications to 200 patients daily. The probability of a medication error per administration is 0.005 (0.5 percent). What is the expected number of errors per day?

$E(\text{errors}) = 200 \times 0.005 = 1$

On average, one error per day is expected. Quality improvement teams use this calculation to set benchmarks and evaluate whether new safety protocols reduce the error rate.

Common Mistakes

1. Adding probabilities for independent events instead of multiplying. “A and B” with independence → multiply. “A or B” with mutual exclusivity → add.
2. Forgetting to subtract the overlap. When events are NOT mutually exclusive, use the general addition rule: $P(A) + P(B) - P(A \text{ and } B)$.
3. Confusing independent and mutually exclusive. Mutually exclusive events cannot both happen (probability of both = 0). Independent events can both happen but do not affect each other.
4. Using permutations instead of combinations. When drawing cards or selecting items, order typically does not matter — use $C(n, r)$, not $P(n, r)$.

Practice Problems

Key Takeaways

• Basic probability = favorable outcomes / total outcomes (for equally likely outcomes)
• Complement rule: $P(E') = 1 - P(E)$ — especially useful for “at least one” problems
• Independent events: $P(A \text{ and } B) = P(A) \times P(B)$ (multiply for “and”)
• Mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B)$ (add for “or”)
• General addition rule: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$ when events can overlap
• Expected value is the probability-weighted average outcome: $E(X) = \sum x_i \cdot P(x_i)$
• Use combinations to count favorable and total outcomes in selection problems

Return to College Algebra for more topics in this section.