Probability Formulas

Why This Matters

Probability formulas are the backbone of quantitative business decision-making—and your exams will test whether you can apply them, not just recite them. You're being tested on your ability to recognize which formula fits a given scenario, whether you're calculating the chance of multiple events occurring, updating beliefs with new information, or measuring the risk associated with an investment. These concepts connect directly to risk assessment, forecasting, expected value analysis, and statistical inference.

Don't just memorize the formulas—know what problem each one solves and when to reach for it. The difference between a strong exam performance and a weak one often comes down to recognizing whether events are independent or dependent, whether you need a complement or a conditional probability, and whether you're dealing with discrete outcomes or continuous distributions. Master the "why" behind each formula, and the calculations will follow naturally.

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Combining Event Probabilities

These formulas help you calculate the probability of multiple events occurring together or separately. The key distinction is whether events can happen simultaneously and whether one event affects another.

Addition Rule of Probability

• Calculates the probability of at least one event occurring—use this when you see "or" in a problem
• Mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B)$—no overlap exists between outcomes
• Non-mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$—subtract the overlap to avoid double-counting

Multiplication Rule of Probability

• Calculates the probability of two events both occurring—use this when you see "and" in a problem
• Independent events: $P(A \text{ and } B) = P(A) \times P(B)$—one event doesn't affect the other
• Dependent events: $P(A \text{ and } B) = P(A) \times P(B|A)$—must account for how the first event changes the second

Probability of Complement

• Calculates the probability an event does NOT occur—often easier than calculating the event directly
• Formula: $P(A') = 1 - P(A)$—especially useful for "at least one" problems
• Strategic shortcut: When asked "what's the probability of at least one success," calculate $1 - P(\text{no successes})$ instead

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Updating Probabilities with Information

These formulas address how probabilities change when you have additional context. Conditional probability is the foundation; Bayes' Theorem is the tool for reversing the conditioning.

Conditional Probability

• Probability of event A given that event B has occurred—written as $P(A|B)$
• Formula: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$—restricts your sample space to only outcomes where B happened
• Business application: Calculating default rates among customers who missed a payment, or conversion rates among visitors who clicked an ad

Bayes' Theorem

• Reverses conditional probabilities—lets you find $P(A|B)$ when you know $P(B|A)$
• Formula: $P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}$—updates your prior belief with new evidence
• Critical for decision-making: Medical testing, spam filtering, and revising sales forecasts as new data arrives

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Measuring Expected Outcomes and Risk

These formulas quantify what you can expect on average and how much variability surrounds that expectation. Expected value tells you the center; variance and standard deviation tell you the spread.

Expected Value

• The long-run average outcome—what you'd expect if you repeated the scenario infinitely
• Formula: $E(X) = \sum [x \times P(x)]$—multiply each outcome by its probability, then sum
• Decision rule: Compare expected values across options; higher $E(X)$ generally means better average payoff

Variance and Standard Deviation

• Variance measures how spread out outcomes are from the mean—formula: $\text{Var}(X) = E[(X - \mu)^2]$ or $E(X^2) - [E(X)]^2$
• Standard deviation is $\sigma = \sqrt{\text{Var}(X)}$—same units as the original variable, easier to interpret
• Risk assessment: Two investments with equal expected returns but different standard deviations carry different risk profiles

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Discrete Probability Distributions

These formulas model specific types of random processes with countable outcomes. Binomial handles fixed trials with two outcomes; Poisson handles event counts over intervals.

Binomial Probability Formula

• Models the number of successes in a fixed number of independent trials—each trial has the same probability $p$
• Formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$—where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$
• Business applications: Quality control (defective items in a batch), sales conversion rates, survey response rates

Poisson Probability Formula

• Models the count of events in a fixed interval—when events occur independently at a constant average rate $\lambda$
• Formula: $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$—where $\lambda$ is both the mean and variance
• Business applications: Customer arrivals per hour, website hits per day, insurance claims per month

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Continuous Probability Distributions

The normal distribution models continuous variables and forms the basis for most statistical inference. Its bell-shaped curve is defined entirely by two parameters.

Normal Distribution

• Characterized by mean ($\mu$) and standard deviation ($\sigma$)—the curve is symmetric around $\mu$
• 68-95-99.7 Rule: Approximately 68% of data falls within $\pm 1\sigma$, 95% within $\pm 2\sigma$, 99.7% within $\pm 3\sigma$
• Foundation for inference: Z-scores, confidence intervals, and hypothesis testing all rely on normal distribution properties

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Quick Reference Table

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Self-Check Questions

1. A problem states that two events cannot occur at the same time. Which formula simplifies, and how does the calculation change compared to non-mutually exclusive events?

2. You know the probability of a positive test result given a disease, but you need the probability of having the disease given a positive test. Which formula do you use, and what additional information do you need?

3. Compare and contrast when you would use the binomial formula versus the Poisson formula. Give one business scenario for each.

4. Two investment options have the same expected value of $10,000. Option A has a standard deviation of $500, and Option B has a standard deviation of $2,000. Which is riskier, and how would you explain this to a client?

5. An FRQ asks: "What is the probability that at least one of five independent machines fails?" Describe the most efficient approach to solve this problem, identifying which formula(s) you would use.