Introduction to the Binomial Distribution and Discrete Probability Models

The study of discrete probability distributions focuses on specific models that define probabilities for random variables.
While there are many types of discrete distributions, this module focuses on three primary ones:
- Binomial Distribution: Used to define probabilities for independent trials.
- Poisson Distribution: Used to define probabilities for events occurring over a specific period of time.
- Hypergeometric Distribution: Used to define probabilities for dependent trials (where trials are not independent).
All discrete probability distributions must obey two fundamental rules:
- The sum of probabilities for all unique events in the sample space must equal 1 ( $\sum P(x) = 1$ ).
- Expected value and variance can be calculated for these distributions. While standard methods apply, specific distributions often have mathematical shortcuts for these calculations.

To identify if a situation follows a binomial distribution, it must meet the following criteria:

Fixed Number of Observations ( $n$ ): There is a set number of trials or observations, denoted by the parameter $n$ .
Binary Outcomes: Each trial results in one of only two possible outcomes, classified as a "success" or a "failure."
- These outcomes must be mutually exclusive (cannot happen at the same time) and collectively exhaustive (nothing else can happen).
- "Success" is defined as the occurrence of the specific event of interest; it does not necessarily imply a positive qualitative outcome.
Constant Probability of Success ( $\pi$ ): The probability of success, denoted by the symbol $\pi$ (not to be confused with the mathematical constant $3.14159...$ ), remains the same from one trial to the next.
Probability of Failure: Since there are only two outcomes, the probability of failure is always equal to $1 - \pi$ .
Independence: Each observation or trial is independent of all others. The outcome of one trial has no effect on the outcome of previous or future trials.
- Counter-Examples (Dependency):
- Weather: If it rained yesterday, there is statistically a higher chance it will rain today. Historical patterns affect future outcomes.
- Stock Prices: Historical fluctuations often have some impact on future price movements, indicating a level of dependency.

The binomial distribution is applicable in various fields where outcomes are binary:

Manufacturing: Labeled items are classified as either "defective" or "acceptable."
Contract Bidding: A firm bidding for a contract either "receives the contract" or "does not receive the contract."
Marketing Research: Survey responses are categorized as "Yes, I will buy" or "No, I will not buy."
Human Resources: Job applicants either "accept the offer" or "reject the offer."

Counting Rule 5 is essential for determining the number of ways a specific number of successes can occur within a set number of trials.

Conceptual Example: In $n = 3$ tosses of a coin, how many ways can you obtain exactly $x = 2$ heads?
- Way 1: Heads, Heads, Tails (HHT)
- Way 2: Heads, Tails, Heads (HTH)
- Way 3: Tails, Heads, Heads (THH)
- There are exactly 3 ways to achieve 2 heads in 3 tosses.
The Combination Formula:
- $\binom{n}{x} = \frac{n!}{x!(n - x)!}$
Notation: In binomial contexts, the notation $\binom{n}{x}$ (long brackets with $n$ over $x$ ) is frequently used as shorthand for combinations ( $nCx$ ). It is not a fraction and does not use a dividing line.

The binomial formula calculates the probability that a random variable $X$ will result in exactly $x$ successes, given the parameters $n$ and $\pi$ .

Formula Structure:
- $P(X = x | n, \pi) = \binom{n}{x} \pi^x (1 - \pi)^{n-x}$
Component Breakdown:
- $n$ : Number of trials.
- $\pi$ : Constant probability of success.
- $x$ : The number of successes of interest.
- $\binom{n}{x}$ : The total number of ways to arrange $x$ successes in $n$ trials.
- $\pi^x$ : The probability of the successes occurring.
- $(1 - \pi)^{n-x}$ : The probability of the failures occurring ( $n - x$ is the number of failures).
Parameters: The values $n$ and $\pi$ are referred to as parameters because they define the specific nature of the distribution.

Scenario: A fair coin is flipped $n = 4$ times. Success is defined as landing on heads.
Parameters: $n = 4$ , $\pi = 0.5$ .
Sample Space for successes ( $x$ ): $\{0, 1, 2, 3, 4\}$ . Note that $x$ cannot exceed $n$ .
Calculation for $P(X = 1)$ :
- $P(X = 1 | 4, 0.5) = \binom{4}{1} (0.5)^1 (0.5)^{4-1}$
- $P(X = 1 | 4, 0.5) = 4 \times 0.5 \times 0.125 = 0.25$

Scenario: Five observations ( $n = 5$ ) with a probability of success $\pi = 0.1$ .
Calculation for $P(X = 1)$ :
- $P(X = 1 | 5, 0.1) = \binom{5}{1} (0.1)^1 (0.9)^4$
- Expanding the logic: This accounts for 1 success ( $0.1$ ) and 4 failures ( $0.9 \times 0.9 \times 0.9 \times 0.9$ ) multiplied by the 5 different positions the success could occupy.
- Result: $0.32805$
Calculation for $P(X = 5)$ :
- $P(X = 5 | 5, 0.1) = \binom{5}{5} (0.1)^5 (0.9)^0$
- Result in scientific notation: $1 \times 10^{-5}$ (or $0.00001$ ).
- Intuition: If an event is very unlikely to happen ( $0.1$ ), the chance of it happening 5 times in a row is extremely low.

The shape of a binomial distribution is directly influenced by its parameters, particularly the probability of success ( $\pi$ ).

The function used is =BINOM.DIST(x, n, pi, cumulative).
- x: Number of successes.
- n: Number of trials.
- pi: Probability of success.
- cumulative: A boolean value. For individual probabilities, use false (or 0).

Low $\pi$ (e.g., $0.1$ ): The distribution is skewed to the right. The highest probability occurs at low values of $x$ (e.g., zero or one success).
Moderate $\pi$ (e.g., $0.5$ ): The distribution is symmetrical and balanced. The probability mass is centered around the middle of the sample space.
High $\pi$ (e.g., $0.9$ ): The distribution is skewed to the left. High probabilities are associated with a high number of successes.

On most Casio calculators, the combination function ( $nCx$ ) is often accessed by pressing the SHIFT key followed by the ÷ (division) key.
While the HP calculator can perform these calculations, it may be less intuitive as it does not display the expression as clearly as modern Casio models.