Study Notes on Hypergeometric Probability and Probability Theory

Permutations versus Combinations

Probability calculations are often built upon the foundation of counting techniques learned in previous lectures: permutations and combinations.

• Permutations (Order Matters):

  • Permutations are used when a specific sequence or order of elements is required.

  • Sports Race Example: Consider a race with $12$ athletes where the first three finishers receive gold, silver, and bronze medals. In this scenario, the order of the winners is critical; a specific trio winning in the order (A, B, C) is a different outcome than (B, A, C) because the medals assigned are unique.

  • Formula for Permutations ($P_{n, r}$):     $P(n, r) = \frac{n!}{(n-r)!}$

  • Evaluating the race example: to distribute $3$ unique medals among $12$ athletes, we calculate:     $\frac{12!}{(12-3)!} = \frac{12!}{9!}$

  • The numerator ($12!$) represents the permutations if all $12$ athletes were given unique medals. We divide by $9!$ (the number of permutations for the remaining athletes who do not receive medals) to isolate the permutations of the prize winners.

• Combinations (Order Does Not Matter):

  • Combinations are used when the selection of a group is important, regardless of the sequence of selection.

  • Team Selection Example: From a group of the top $5$ athletes, $2$ are randomly selected for a national team. Selecting person A then person B results in the same team as selecting person B then person A.

  • Formula for Combinations ($C_{n, r}$ or "n choose r"):     $\binom{n}{r} = \frac{n!}{r!(n-r)!}$

  • This formula includes an extra division by $r!$ compared to permutations because the number of internal sequences among those chosen does not matter; they all form a single group.

Hypergeometric Probability: Conceptual Framework

Hypergeometric probability is a specific calculation determining the likelihood of certain outcomes when drawing from a population without replacement.

• Drawing Without Replacement: This is a defining characteristic. Once an item (or person) is selected from the pool, they are not returned. This means they cannot be selected twice in the same sample.

• Success and Failure Vocabulary: In mathematics, specific outcomes of interest are termed "successes" and others "failures." This nomenclature is strictly technical and carries no moral or qualitative judgment.

The Smoker Case Study: Step-by-Step Calculation

Consider a small population of $9$ persons: $4$ smokers and $5$ non-smokers. A sample of $3$ persons is chosen randomly. The goal is to find the probability that exactly $1$ smoker and $2$ non-smokers are chosen.

• Step 1: Total possible samples. Determine the total ways to choose $3$ people from $9$. Since order does not matter, we use combinations:     $\binom{9}{3} = \frac{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1) \times (6 \times 5 \times 4 \times 3 \times 2 \times 1)} = 84$     There are $84$ possible combinations.

• Step 2: Probability of a single sample. Assuming no bias (every person has an equal chance of selection), every one of the $84$ possible samples has an equal probability:     $P(\text{each sample}) = \frac{1}{84}$

• Step 3: Calculating target combinations. Find how many ways to pick specifically $1$ smoker and $2$ non-smokers.

  • Ways to choose $1$ smoker from $4$ available: $\binom{4}{1} = 4$

  • Ways to choose $2$ non-smokers from $5$ available: $\binom{5}{2} = 10$

  • The total number of desired combinations is the product of these two (the "and" rule for combinations):     $4 \times 10 = 40$

• Step 4: Final Probability. Multiply the number of desired combinations by the probability of each sample:     $P(X=1) = 40 \times \frac{1}{84} \thickapprox 0.476$     This is equivalent to $47.6\%$.

Generalizing the Hypergeometric Formula

To apply this to any scenario involving drawing without replacement, let:

• $N$ = Total Population size.

• $k$ = Total number of successes available in the population.

• $N - k$ = Total number of failures in the population.

• $n$ = Sample size.

• $x$ = Number of successes in the sample.

• $n - x$ = Number of failures in the sample.

The Hypergeometric Formula: $P(X=x) = \frac{\binom{k}{x} \binom{N-k}{n-x}}{\binom{N}{n}}$

• Structural Consistency: In the numerator, the sum of the top symbols in the combinations ($k + (N-k)$) equals the total population $N$. The sum of the bottom symbols ($x + (n-x)$) equals the sample size $n$. This logic can be extended to populations with more than two categories of items.

Probability Tree Representation

A probability tree can visualize the smoker example by mapping outcomes branch by branch (first person, second person, third person).

• Branching and Dependencies: Because there is no replacement, the probabilities change at each step. If the first person chosen is a smoker (probability $4/9$), there are only $3$ smokers left out of $8$ remaining people for the second draw.

• Multiplication Rule (AND): When moving from left to right along a branch (e.g., Smoker AND Non-Smoker AND Non-Smoker), multiply the probabilities.     $P(S \text{ and } NS \text{ and } NS) = \frac{4}{9} \times \frac{5}{8} \times \frac{4}{7}$

• Addition Rule (OR): To find the total probability of an outcome that can occur via different sequences (Smoker first, Smoker second, or Smoker third), calculate the probability for each path and add them together. Since each sequence has the same numerical outcome, the result is $3 \times 0.1587... = 0.476$.

Works Council Case Study: Complex Hypergeometric Application

Scenario Details: A company has $80$ employees.

• $50$ are full-time (FT).

• $30$ are part-time (PT).

• $36$ are junior workers ($18$ FT and $18$ PT).

• $44$ are senior workers ($32$ FT and $12$ PT).

• A council of $5$ members is chosen at random without replacement.

Calculating Exactly One Part-Timer: Using the hypergeometric formula, we distinguish between PT (success) and FT (failure).

• Sample size ($n$) = $5$

• Population size ($N$) = $80$

• Total PT ($k$) = $30$

• Total FT ($N-k$) = $50$

• Desired PT in sample ($x$) = $1$

• Desired FT in sample ($n-x$) = $4$

$P(\text{Exactly 1 PT}) = \frac{\binom{30}{1} \binom{50}{4}}{\binom{80}{5}}$

Calculating At Least One Part-Timer (The Complementary Rule): Calculating at least one involves summing the probabilities of having $1, 2, 3, 4, \text{ or } 5$ part-timers. This is tedious and prone to error.

• The Complementary Rule: The sum of all possible outcomes equals $1$ ($100\%$). Therefore, the probability of at least one is $1$ minus the probability of zero.     $P(\text{at least 1 PT}) = 1 - P(0 \text{ PT})$

• In this calculation, we look for samples of $5$ where all members are FT:     $P(0 \text{ PT}) = \frac{\binom{30}{0} \binom{50}{5}}{\binom{80}{5}}$

Fundamental Concepts of Probability Theory

• Random Experiment: A process where the outcome is uncertain but can be repeated under identical circumstances (e.g., flipping a coin, rolling a die).

• Outcome Space (Sample Space $S$): A finite or infinite set containing all possible simple events. For flipping a coin twice: $S = \{HH, HT, TH, TT\}$.

• Event: A subset of the outcome space.

  • Simple Event: A single outcome (e.g., getting exactly Heads and then Tails).

  • Event (Complex): A collection of outcomes (e.g., "at least one tails" in two flips includes $HT, TH, \text{ and } TT$).

The Three Definitions of Probability

1. Classical Definition (Laplace): Based on theoretical symmetry. The probability of event A is the number of simple events in A divided by the total number of simple events in the sample space. This assumes all outcomes are equally likely (e.g., a fair die).

2. Relative Frequency Definition: Based on observation. The probability is estimated by repeating an experiment many times (e.g., $10,000$ times) and determining the percentage of times the event occurs. This is used if a die or coin is biased.

3. Subjective Definition: Based on prognosis or expert opinion for events that cannot be repeated (e.g., presidential election outcomes). Probability may vary between different observers or surveys.

Core Rules of Probability

• Rule 1 (Range): Probabilities must always fall between $0$ and $1$.

  • $P(A) = 0$ means the event never happens.

  • $P(A) = 1$ means the event always happens.

  • Negative probabilities or values greater than $1$ are mathematically impossible and indicate calculation errors.

• Rule 2 (Total Probability): The sum of probabilities for all possible mutually exclusive outcomes in the sample space must equal $1$.     $P(S) = 1$

Questions & Discussion

• Question: What happens if the selection is done with replacement?

• Answer: If items are replaced after being chosen (e.g., selecting the same person for a council twice or choosing the same multiple-choice answer across Different questions), the probabilities remain constant for each draw. This scenario follows a Binomial Probability distribution, which will be discussed in the next lecture.

• Question: Is the classical formula ($\frac{\text{elements in A}}{\text{elements in S}}$) practical for exams?

• Answer: Yes, it is used in specific contexts. For example, when you determine that a fair die has a $1/6$ probability for each side, you are using the classical definition. This is fundamentally different from the hypergeometric scenario, which deals with populations of differing categories.