Probability Basics

Probability is a measure of how likely an event is to happen. It is expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain.

$P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

A probability of 0.5 (or 50%) means the event is equally likely to happen or not happen — like flipping a fair coin.

Basic Probability

Example 1: Rolling a Die

A standard die has 6 faces numbered 1 through 6. What is the probability of rolling a 4?

There is 1 favorable outcome (rolling a 4) and 6 total possible outcomes.

$P(4) = \frac{1}{6} \approx 0.167 = 16.7\%$

Answer: The probability is $\frac{1}{6}$, or about 16.7%.

Example 2: Drawing a Card

A standard deck has 52 cards. What is the probability of drawing a heart?

There are 13 hearts in a deck of 52 cards.

$P(\text{heart}) = \frac{13}{52} = \frac{1}{4} = 0.25 = 25\%$

Answer: The probability is $\frac{1}{4}$, or 25%.

Converting Probability to a Percentage

To convert any probability to a percentage, multiply by 100:

$P = 0.35 \Rightarrow 0.35 \times 100 = 35\%$

The Complement Rule

The complement of an event is everything that is not that event. The probability of an event not happening is:

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

This is useful when it is easier to calculate what you don’t want.

Example 3: Probability of NOT Rolling a 6

$P(\text{not } 6) = 1 - P(6) = 1 - \frac{1}{6} = \frac{5}{6} \approx 83.3\%$

Answer: There is about an 83.3% chance of rolling something other than a 6.

Example 4: Product Defects

A factory produces 200 items per day, and on average 8 are defective. What is the probability that a randomly selected item is NOT defective?

$P(\text{defective}) = \frac{8}{200} = 0.04$

$P(\text{not defective}) = 1 - 0.04 = 0.96 = 96\%$

Answer: There is a 96% chance the item is not defective.

Independent Events (AND)

Two events are independent if the outcome of one does not affect the outcome of the other. To find the probability that both events happen, multiply their probabilities:

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

Example 5: Flipping Two Coins

What is the probability of getting heads on both flips?

$P(\text{heads and heads}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} = 25\%$

Answer: The probability is 25%.

Example 6: Two Independent Inspections

A product passes through two independent quality checks. The probability of passing the first is 0.95 and the second is 0.90. What is the probability of passing both?

$P(\text{pass both}) = 0.95 \times 0.90 = 0.855 = 85.5\%$

Answer: There is an 85.5% chance the product passes both inspections.

Mutually Exclusive Events (OR)

Two events are mutually exclusive if they cannot happen at the same time. To find the probability that either one happens, add their probabilities:

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

Example 7: Rolling a Die

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

$P(2 \text{ or } 5) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \approx 33.3\%$

Answer: The probability is $\frac{1}{3}$, or about 33.3%.

Important: This rule only works when the events cannot overlap. If events can overlap (they are not mutually exclusive), you must subtract the overlap:

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

Probability Rules Reference

The Venn diagram above shows a standard deck of 52 cards. There are 26 red cards, 12 face cards, and 6 cards that are both red and face cards. Using the general addition rule: $P(\text{red or face}) = \frac{26}{52} + \frac{12}{52} - \frac{6}{52} = \frac{32}{52} = \frac{8}{13}$.

Real-World Application: Nursing — Interpreting Diagnostic Test Results

A hospital uses a rapid screening test for a certain condition. Based on historical data:

• The condition occurs in 5% of the tested population: $P(\text{condition}) = 0.05$
• The test correctly identifies patients with the condition 92% of the time (sensitivity): $P(\text{positive} \mid \text{condition}) = 0.92$
• The test correctly identifies patients without the condition 88% of the time (specificity): $P(\text{negative} \mid \text{no condition}) = 0.88$

Question: If 200 patients are tested, how many do we expect to test positive who actually have the condition (true positives)?

Step 1: Find how many patients likely have the condition.

$200 \times 0.05 = 10 \text{ patients}$

Step 2: Of those 10, the test identifies 92% correctly.

$10 \times 0.92 = 9.2 \approx 9 \text{ true positives}$

Step 3: How many patients without the condition will get a false positive?

$200 - 10 = 190 \text{ patients without the condition}$

$190 \times (1 - 0.88) = 190 \times 0.12 = 22.8 \approx 23 \text{ false positives}$

Interpretation: Of the roughly $9 + 23 = 32$ total positive results, only about 9 are true positives. This means that even with a positive test result, there is roughly a $\frac{9}{32} \approx 28\%$ chance the patient actually has the condition. This is why nurses and doctors use follow-up testing rather than relying on a single screening result.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Next Steps

This page covers the fundamentals. The probability cluster continues with more advanced topics:

• Counting Techniques — permutations and combinations for calculating total outcomes
• Addition Rule — the full addition rule with Venn diagrams
• Compound Events and Independence — multiplication rule for dependent events, with and without replacement
• Conditional Probability — P(A|B) from formulas and two-way tables
• Bayes’ Theorem — reversing conditional probabilities for medical testing and more

Key Takeaways

• Probability ranges from 0 (impossible) to 1 (certain). Multiply by 100 to convert to a percentage.
• Basic formula: favorable outcomes divided by total outcomes.
• Complement rule: $P(\text{not } A) = 1 - P(A)$ — often the easiest way to solve a problem.
• AND (independent events): multiply the probabilities.
• OR (mutually exclusive events): add the probabilities.
• In real-world applications like medical testing, understanding probability helps you interpret results correctly — a positive screening test does not always mean the condition is present.

Return to Statistics for more topics in this section.