Introduction to Probability and Discrete Distributions

Addition Rules: Mutually Exclusive vs. Overlap

When we want to find the probability of one event or another occurring, we use the Addition Rule.

1. Mutually Exclusive Events: If the events cannot overlap, simply add their probabilities. $P(A \text{ or } B) = P(A) + P(B)$ Example: If $P(\text{Mobile}) = 0.42$ and $P(\text{Desktop}) = 0.33$, then $P(\text{Mobile or Desktop}) = 0.75$.

2. Events with Overlap: If the events can happen simultaneously, adding them directly would count the overlap twice. We must subtract the joint probability once: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Joint and Conditional Probability

Understanding the relationship between two variables requires distinguishing between joint and conditional probabilities.

• Joint Probability: This is the probability that two events happen at the same time, often represented as $P(A \text{ and } B)$. In a table of 300 shipments, if 12 were both "Air" and "Damaged," the joint probability is $\frac{12}{300} = 0.04$.
• Conditional Probability: This is the probability of an event occurring given that another event has already occurred. The wording "among [group]" often signals a conditional probability. $P(A | B) = \frac{P(A \text{ and } B)}{P(B)}$ If we want to know the probability a package is damaged given it was sent via Air, we use the total number of Air shipments as our denominator.
• Reverse Conditional Probability: This involves switching the condition. For example, instead of $P(\text{Damaged} | \text{Air})$, we might ask for $P(\text{Air} | \text{Damaged})$. Here, the denominator becomes the total number of damaged packages.

Exam Tip: The Independence Check To check if two events are independent, compare the conditional probability to the marginal probability. If $P(A | B) = P(A)$, the events are independent. If they are not equal, the events are dependent.

Discrete Probability Distributions

A discrete variable is one that has separate, countable values, such as the number of customer returns. This is distinct from continuous variables, such as weight or temperature, which can take any value within a range.

Valid Distributions

For a discrete probability distribution to be valid, it must meet two criteria:

1. Every individual probability $P(x)$ must be between 0 and 1.
2. The sum of all probabilities must equal exactly 1.

Expected Value (The Long-Run Average)

The Expected Value, denoted as $E(X)$, is a probability-weighted mean. It represents the long-run average value we would expect if we repeated the experiment many times. To calculate it, multiply each possible value $x$ by its corresponding probability $P(x)$ and sum the results: $E(X) = \sum x \cdot P(x)$

If a distribution has values $x = {0, 1, 2, 3}$ with probabilities ${0.15, 0.35, 0.30, 0.20}$, the calculation is: $E(X) = (0 \cdot 0.15) + (1 \cdot 0.35) + (2 \cdot 0.30) + (3 \cdot 0.20) = 1.55$ .

Standard Deviation and Risk

While the expected value tells us the average, the standard deviation tells us how much the actual outcomes vary or "spread" around that average. It is expressed in the same units as the original variable.

In business contexts, standard deviation is a proxy for risk.

In the table above, Plan B has a higher expected profit, but its much larger standard deviation indicates significantly higher uncertainty and risk compared to Plan A.