Case Analysis

ABSTRACT

Case analysis is a problem-solving strategy used to calculate probabilities or expected values by partitioning a complex problem into smaller, mutually exclusive, and collectively exhaustive scenarios (cases).

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The Core Concept

Case analysis relies on the Law of Total Probability and the Law of Total Expectation. By breaking a sample space $S$ into disjoint cases $C_1,C_2,\dots ,C_n$, you can solve for each case individually and then aggregate the results.

Conditions for Cases

To ensure the analysis is mathematically sound, the chosen cases must be:

• Mutually Exclusive: No two cases can happen at the same time ($C_i\cap C_j=\varnothing$).
• Collectively Exhaustive: The cases must cover every possible outcome of the experiment ($\bigcup C_i=S$).

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Probability by Case Analysis

When calculating the probability of an event $A$, you can use the Law of Total Probability:

$$P(A)=\sum _{i=1}^{n}P(A|C_i)P(C_i)$$

Example: Selecting a Coin

Suppose you have two jars. Jar 1 has 3 gold coins and 1 silver coin. Jar 2 has 1 gold coin and 3 silver coins. You pick a jar at random (50% chance each) and then pick a coin. What is the probability of picking a gold coin ($G$)?

• Case 1 ($C_1$): You pick Jar 1. $P(C_1)=0.5$, $P(G|C_1)=0.75$.
• Case 2 ($C_2$): You pick Jar 2. $P(C_2)=0.5$, $P(G|C_2)=0.25$.
• Result: $P(G)=(0.75\cdot 0.5)+(0.25\cdot 0.5)=0.5$.

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Expectation by Case Analysis

Similarly, the Law of Total Expectation allows you to calculate the expected value $E[X]$ by weighting the expectation within each case by the probability of that case occurring:

$$E[X]=\sum _{i=1}^{n}E[X|C_i)P(C_i)$$

This is particularly useful for problems involving Recursive Algorithms or state-based transitions, such as finding the expected number of trials in a Geometric Distribution.

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When to Use Case Analysis

Scenario	Application
Multi-stage processes	Use cases to represent different paths in a decision tree.
Unknown Parameters	Use cases when the probability of an event depends on an unknown prior condition.
Complex Sample Spaces	Use cases to simplify “at least” or “at most” problems by looking at disjoint counts.

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Related Notes

• Conditional Probabilities: The foundation of the “if-then” logic used in defining cases.
• Conditional Expectation: The mathematical formalization of $E[X|C_i]$.
• Bayes’ Theorem: Often used after a case analysis to “reverse” the condition (e.g., given we found a gold coin, what is the probability it came from Jar 1?).
• Geometric Distribution: Frequently solved using case analysis on the first trial (Success vs. Failure).