Probability and Conditional Probability for the TMUA

Combining Probabilities: Addition and Multiplication

Probability is measured on a scale from 0 to 1, where 0 represents impossibility and 1 represents certainty. When dealing with multiple events, you must determine whether to add or multiply their probabilities based on the relationship between them.

1. The Addition Rule (OR logic): You add probabilities when you want to find the likelihood of at least one of two events occurring. If events $A$ and $B$ are mutually exclusive, meaning they cannot happen at the same time, the probability of $A$ or $B$ occurring is $P(A \cup B) = P(A) + P(B)$. If they are not mutually exclusive, you must subtract the intersection to avoid double counting: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

2. The Multiplication Rule (AND logic): You multiply probabilities when you want to find the likelihood of both events occurring. For independent events, where the outcome of one does not affect the other, $P(A \cap B) = P(A) \times P(B)$. If the events are dependent, you must use the conditional probability: $P(A \cap B) = P(A) \times P(B|A)$.

Understanding Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. This is written as $P(A|B)$, read as the probability of $A$ given $B$. The formal definition is:

$P(A|B) = \frac{P(A \cap B)}{P(B)}$

This formula implies that $P(A \cap B) = P(B)P(A|B)$. In a conditional probability problem, the sample space is restricted to only those outcomes where $B$ is true. If $A$ and $B$ are independent, then $P(A|B) = P(A)$, as the occurrence of $B$ provides no new information about the likelihood of $A$.

Using Venn Diagrams and Two-Way Tables

Venn diagrams and two-way tables are powerful tools for interpreting intersections and conditional probabilities. A Venn diagram represents events as overlapping circles within a universal set. The overlapping region represents $P(A \cap B)$, while the area inside either circle represents $P(A \cup B)$.

Two-way tables allow for the representation of probabilities using expected frequencies. This involves assuming a large total population, such as 1000, and filling the cells with the number of individuals expected to fall into each category.

Worked Example: Two-Way Table Suppose a medical test is 95 percent accurate for a disease that affects 1 in 100 people. If we take a sample of 10,000 people:

1. Expected with disease: $0.01 \times 10,000 = 100$.
2. Expected without disease: 9,900.
3. Disease and Test Positive (Correct): $0.95 \times 100 = 95$.
4. Disease and Test Negative (Error): 5.
5. No Disease and Test Negative (Correct): $0.95 \times 9,900 = 9,405$.
6. No Disease and Test Positive (Error): $0.05 \times 9,900 = 495$.

From the table, the probability that someone has the disease given they tested positive is $P(D|Pos) = \frac{95}{95 + 495} = \frac{95}{590} \approx 0.161$. This shows that even with a 95 percent accurate test, the conditional probability of having a rare disease after a positive result can be surprisingly low.

Tree Diagrams for Combined Events

Tree diagrams represent sequential events. Each set of branches must sum to a probability of 1. To find the probability of a specific path, multiply the probabilities along the branches. To find the total probability of an outcome that occurs on multiple paths, sum the probabilities of those paths.

a. Independent Outcomes: When probabilities are independent of previous outcomes (such as flipping a coin or drawing with replacement), the probabilities on the second set of branches are identical to the first.

b. Dependent Outcomes: When probabilities are dependent on previous outcomes (such as drawing cards from a deck without replacement), the probabilities on subsequent branches must be updated. For example, if a bag contains 5 red and 3 blue balls, the probability of picking red first is $5/8$. If a red ball is removed, the probability of picking red second becomes $4/7$. The tree diagram clearly illustrates $P(R_2|R_1)$ on the second set of branches.