Binomial Distribution and Proportions

Binomial Distribution: A discrete probability distribution that describes the number of successes in a sequence of independent experiments or trials.
Key Learning Objectives:
- Understand the difference between probabilities and proportions.
- Distinguish between parameter and estimate.
- Explain the binomial distribution clearly.
- Connect binomial distribution concepts to basic probability rules.
- Quantify expected mean and variability in outcomes.
- Conduct and interpret a binomial test effectively.

Definition: A probability refers to the likelihood of a particular outcome occurring, while a proportion indicates the fraction of the total outcomes that exhibit a certain characteristic.
Example: If 6101 out of 9821 slices of toast landed butter side down, the probability of landing butter side down is based on these observed results, giving a proportion of 6101/9821.

True Probability Assumption: Assume the true probability of butter side down is 60% (p = 0.6).
Main Query: When three people drop their slices of toast, what is the probability that one slice lands butter side down and two slices land butter side up?
Method to Approach: Draw a probability tree to help visualize outcomes and their respective probabilities.

Bernoulli Trial (BT): A random experiment with exactly two possible outcomes (success or failure).
Examples:
- Question: Is the top card of a shuffled deck an Ace? (Yes/No)
- Question: Was the newborn child a girl? (Yes/No)

Definition: The binomial distribution gives the probability of obtaining a specific number of successes (X) in a fixed number (n) of independent trials.
Parameters:
- n: Total number of trials.
- p: Probability of success on each trial.
Assumptions:
- Trials are independent (outcomes do not affect each other).
- Each trial has two outcomes (success or failure).

Define Success and Failure: Identify what constitutes a 'success' (e.g., slice landing butter side down).
Set Up the Binomial Coefficient: Denoted as “n choose X” and can be calculated using the formula: inom{n}{X} = \frac{n!}{X!(n-X)!}
- Example: If flipping n coins, how many heads do we get?
Probability of Specific Arrangement: Calculate based on defined success and failure outcomes.
Combine Binomial Coefficient with Probabilities of Success: Determine the product to find total probabilities.

Definition: A way to quantify the number of possible combinations of successes in trials using factorials.
Example Scenario: Given 5 toys (3 axolotls and 2 Labubus), to determine how many ways to choose 2 out of 5.

Variance (σ²): Quantifies how much the probabilities can vary.
ext{Variance: } \sigma^2 = p(1 - p)
where $p$ is the probability of success and $q = 1 - p$.
Standard Deviation (σ): Gives an idea of the spread of probabilities.
ext{Standard Deviation: } \sigma = \sqrt{p(1 - p)}

Null Hypothesis (H0): The sample proportion is derived from a population where the probability of success is equal to p.
Alternate Hypothesis (H1): The sample proportion comes from a population where the probability of success is not equal to p.

Context: Researchers observed that out of 9821 slices of toast, 6101 landed butter side down.
Objective: Test whether this outcome aligns with a 50:50 chance distribution (a typical experimental design question).

The p-value measures the strength of evidence against the null hypothesis. A very small p-value (e.g., $1.05 \times 10^{-128}$) suggests a rejection of the null hypothesis, indicating significant anomaly in results.

If the probability of a toast landing butter side down is 0.6, calculate the expected proportion over many trials.
Define success in a binomial experiment and provide one example.
Calculate the probability of getting exactly one head when flipping a fair coin twice.

Calculate the probability of a basketball player making 3 out of 4 free throw shots if the probability of hitting a free throw is 0.75.
What is the probability of rolling exactly 4 skulls if a die has one skull face and five blank faces, during 20 rolls?
If a newborn has a 0.51 probability of being male, calculate the probability of exactly 6 out of 10 births being female.

Test if the probability of drawing an Ace differs significantly from $\frac{4}{52}$ after 100 draws yielding 12 Aces.
Analyze data from Mars where out of 2051 slices of toast, 938 landed butter side down, to assess adherence to Earth-style probability distribution using binomial testing.