Evidence-Based Approaches to Public Health: Biostatistics – Probability: Basic Probability Concepts
In this tutorial, we will explore basic probability concepts, which are foundational to understanding statistical methods in public health research. Probability helps quantify the likelihood of an event occurring and is critical in making predictions based on data. Mastering these concepts is essential for public health professionals and will be covered on the Certified in Public Health (CPH) exam.
By the end of this tutorial, you will understand the basic rules of probability, key terms, and how to calculate probabilities in different scenarios. Practice questions are included to reinforce your understanding.
Table of Contents:
- Introduction to Probability
- Basic Probability Terms
- Experiment, Sample Space, Event
- Mutually Exclusive Events
- Independent Events
- Rules of Probability
- Addition Rule for Mutually Exclusive Events
- Multiplication Rule for Independent Events
- Complement Rule
- Practice Questions
- Conclusion
1. Introduction to Probability
Probability is the measure of how likely an event is to occur. It ranges from 0 to 1, where 0 means the event is impossible and 1 means the event is certain to happen. In public health research, probability is used to assess risks, predict outcomes, and analyze the likelihood of health events based on data.
The formula for calculating probability is:
[math] P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} [/math]
Where P(A) is the probability of event A occurring.
2. Basic Probability Terms
Before diving into the rules of probability, it’s important to understand some key terms:
2.1 Experiment, Sample Space, Event
- Experiment: Any process that leads to a result. For example, rolling a die or selecting a person at random from a population.
- Sample Space (S): The set of all possible outcomes of an experiment. For example, when rolling a die, the sample space is {1, 2, 3, 4, 5, 6}.
- Event: A specific outcome or a set of outcomes from the sample space. For example, rolling a 3 is an event, and so is rolling an even number.
2.2 Mutually Exclusive Events
Mutually exclusive events are events that cannot happen at the same time. If one event occurs, the other cannot.
- Example: In rolling a die, getting a 2 and getting a 5 are mutually exclusive events because you cannot roll both a 2 and a 5 on a single roll.
2.3 Independent Events
Independent events are events where the outcome of one event does not affect the outcome of the other.
- Example: Flipping a coin and rolling a die are independent events because the result of the coin flip does not influence the result of the die roll.
3. Rules of Probability
There are several key rules of probability that help us calculate the likelihood of events occurring, whether they are mutually exclusive, independent, or otherwise related.
3.1 Addition Rule for Mutually Exclusive Events
If two events are mutually exclusive (i.e., they cannot happen at the same time), the probability that either event occurs is the sum of their individual probabilities.
The formula is:
[math] P(A \text{ or } B) = P(A) + P(B) [/math]
Example: If the probability of rolling a 3 on a die is 1/6, and the probability of rolling a 5 is also 1/6, then the probability of rolling either a 3 or a 5 is:
[math] P(3 \text{ or } 5) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3} [/math]
3.2 Multiplication Rule for Independent Events
If two events are independent, the probability that both events occur is the product of their individual probabilities.
The formula is:
[math] P(A \text{ and } B) = P(A) \times P(B) [/math]
Example: If the probability of flipping heads on a coin is 1/2 and the probability of rolling a 6 on a die is 1/6, the probability of flipping heads and rolling a 6 is:
[math] P(\text{Heads and 6}) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12} [/math]
3.3 Complement Rule
The complement rule is used to find the probability that an event does not occur. The complement of event A is denoted by A’, and the sum of the probabilities of A and A’ is always 1.
The formula is:
[math] P(A’) = 1 – P(A) [/math]
Example: If the probability of it raining tomorrow is 0.30, the probability that it will not rain is:
[math] P(\text{No Rain}) = 1 – 0.30 = 0.70 [/math]
4. Practice Questions
Test your understanding of basic probability concepts with these practice questions. Try answering them before checking the solutions.
Question 1:
A die is rolled. What is the probability of rolling a 4 or a 6?
Answer 1:
Answer, click to reveal
These are mutually exclusive events, so we use the addition rule:
[math] P(4 \text{ or } 6) = P(4) + P(6) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3} [/math]
Question 2:
A coin is flipped and a die is rolled. What is the probability of flipping heads and rolling an odd number?
Answer 2:
Answer, click to reveal
These are independent events, so we use the multiplication rule:
[math] P(\text{Heads and Odd}) = P(\text{Heads}) \times P(\text{Odd}) = \frac{1}{2} \times \frac{3}{6} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} [/math]
Question 3:
The probability that a person selected at random has a certain disease is 0.02. What is the probability that a randomly selected person does not have the disease?
Answer 3:
Answer, click to reveal
We use the complement rule:
[math] P(\text{No Disease}) = 1 – P(\text{Disease}) = 1 – 0.02 = 0.98 [/math]
5. Conclusion
Basic probability concepts are essential for understanding the likelihood of events occurring in public health research. These concepts help researchers make predictions, assess risks, and interpret data in a meaningful way.
Always remember:
- The addition rule is used for mutually exclusive events to find the probability that one event or another occurs.
- The multiplication rule is used for independent events to find the probability that both events occur.
- The complement rule is used to find the probability that an event does not occur.
Final Tip for the CPH Exam:
Make sure you understand how to apply basic probability rules and calculate probabilities for different types of events. Practice solving problems involving mutually exclusive events, independent events, and complements, as these concepts are likely to be tested on the Certified in Public Health (CPH) exam. Additionally, and perhaps more importantly, learn to determine what category a set of events might belong to so as to apply the right rule at the right time.
Humanities Moment
For this CPH Focus, the featured image is Czardas dancers by Ernst Ludwig Kirchner (German, 1880-1938). Kirchner was a pioneering German expressionist and founding member of Die Brücke, a movement that broke from academic tradition to embrace raw emotion, bold color, and modern forms. After early success, World War I and mental health struggles shaped both his life and art, leading to periods of isolation and prolific output in Davos, Switzerland. Condemned as “degenerate” by the Nazis, Kirchner’s later years were marked by political disillusionment and deteriorating health, ending with his death in 1938. He left behind a body of work that helped define early 20th-century modernism.