Study Notes on Binomial Distribution

Section 6.2: The Binomial Distribution

Definition of Binomial Distribution

A binomial distribution is a special case of a discrete probability distribution.
It involves two possible outcomes: "success" and "failure".
It is important not to get bogged down with the specific terminology when approaching this topic.

Requirements for a Binomial Distribution

Fixed Number of Trials (n): The experiment must be performed a specific number of times, denoted as n.
Trials: Each repetition of the experiment is referred to as a trial. There are n trials in total.
Independence: The trials must be independent, meaning the outcome of one trial does not affect the outcomes of the other trials.
Disjoint Outcomes: For each trial, there are only two possible outcomes: success or failure. These events must be "disjoint" or "mutually exclusive," meaning they cannot occur simultaneously.
Constant Probability: The probability of success (denoted as p) remains constant across all trials, while the probability of failure is represented as (1 - p).

Notation in Binomial Distribution

Random Variable (X): If X is the random variable representing the number of successes in the n trials, then X is classified as a binomial random variable.
Number of Trials (n): Denoted as n.
Probability of Success (p): The probability of success for each trial is denoted as p.
Probability of Failure: The probability of failure for each trial is defined as (1 - p).
Range of X: The values of X must fall within the range
$\text{0} \leq X \leq n$ ,
indicating that X can take any integer value between 0 and n, inclusive.

Examples of Binomial Experiments

It is important to determine whether certain scenarios represent binomial experiments:
- Asking 100 people if they like strawberry ice cream: Binomial experiment (two outcomes: like or not like).
- Asking 100 people if they prefer vanilla, chocolate, or strawberry ice cream: Not a binomial experiment (more than two outcomes).
- Flipping a coin 10 times and observing the number of tails: Binomial experiment (two outcomes: heads or tails).
- Rolling a die 10 times and recording the result: Not a binomial experiment (more than two outcomes).
- Rolling a die 10 times and observing the number of 5s: Binomial experiment (two outcomes: 5 or not 5).
- Selecting 5 cards from a deck without replacement and recording the number of hearts: Not a binomial experiment (since the trials are not independent).

Calculating Binomial Probabilities

Methods of Calculation Include:
- By hand using formulas.
- Using StatCrunch (computational software).
- Reference to binomial tables for simpler calculations.

Binomial Probability Formula

To calculate the probability of obtaining exactly x successes in n trials, we use the formula:
$P(X = x) = nC_x \times p^x \times (1-p)^{n-x}$
where nCx is the number of combinations of n trials taken x at a time.

Example of Binomial Probability Calculation

Scenario: Suppose 20% of students graduate with honors. We randomly select 8 graduates. What is the probability that exactly 3 of these graduates have honors?
Parameters:
- n = 8 (number of trials)
- p = 0.20 (probability of success)
- x = 3 (number of successes)
Step-by-step Calculation:
1. Identify parameters:
  - $n = 8$
  - $x = 3$
  - $p = 0.20$
  - $(1-p) = 1 - 0.20 = 0.80$
2. Calculate the number of combinations ( $nC_x$ ):
  - $8C_3 = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!} = \frac{8 \times 7 \times 6 \times 5!}{3 \times 2 \times 1 \times 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$
3. Calculate $p^x$ :
  - $(0.20)^3 = 0.008$
4. Calculate $(1-p)^{n-x}$ :
  - $(0.80)^{8-3} = (0.80)^5 = 0.32768$
5. Multiply the results:
  - $P(X=3) = 56 \times 0.008 \times 0.32768 = 0.14680064$
Use the probability formula to find P(X=3).

Beyond Point Probabilities

In many cases, we are interested in the probability that X equals a specific value. However, more often, we need to explore an interval for X.
To achieve this, we employ the cumulative distribution function (CDF):
- This involves summing the point probabilities of interest from the specified interval.
- It is also beneficial to take advantage of the complement rule:
- P(X > a) = 1 - P(X < a)
- $P(X \text{ is at least 1}) = 1 - P(X = 0)$

Using Binomial Tables

Binomial tables simplify calculations made before the advent of calculators and software.
To find a specific probability, locate the appropriate:
- Section for your n.
- Row for x and column for p to find P(X = x).
Some texts, including the provided tables, may use r for x in the tables.
These tables can also be utilized for cumulative calculations (add multiple point probabilities).
Note: Only point probability tables may be available during tests.

Example of Using Binomial Tables

In the previous example, we looked for P(X = 3) when n = 8 and p = 0.20.
Use the section for n = 8, column for p = 0.20, and find r = 3 in the table.

Binomial Probabilities in StatCrunch

To compute binomial probabilities using StatCrunch, follow these steps:
1. Select Stat.
2. Highlight Calculators and choose Binomial.
3. Enter the number of trials (n) and the probability of success (p).
4. Use the standard tab to calculate exact probabilities, including options for <, >, and specific value calculations, by filling in the signs and values for x.
5. Utilize the between tab to calculate the probability of x falling between two values.
6. Click Compute!
Verification: Check the earlier example by finding P(X=3) when n=8 and p=0.2 and question what the probability is that there are at least 3 students with honors or fewer than 2.

Graphing Binomial Distributions

One can graph a binomial probability distribution similarly to any other discrete probability distribution, as described in Section 6.1.
To graph:
1. Obtain the table form of the distribution.
2. Use the graphing approach outlined in Section 6.1.
Note: StatCrunch provides a probability distribution graph in its binomial calculator, which serves as a visual reference.

Binomial Practice Problems

Problem 1

List the 4 requirements of a binomial distribution.
Given that 30% of students in BIO-101 plan to major in science, determine the probabilities for:
- Exactly 2 students planning to major in science.
- No students planning to major in science.
- At least one student planning to major in science.
- Evaluate whether any of these events can be classified as unusual.

Problem 2

Speed is cited as a contributing factor in 72% of fatal traffic accidents. For a random sample of 60 reports from fatal accidents, find:
- The probability that speed was a contributing factor in more than 48 of these accidents.
- Evaluate if this scenario would be considered unusual.
- Calculate the mean of this distribution.
- Compute the standard deviation of the distribution.

Mean and Standard Deviation for Binomial RVs

The mean (µx) and standard deviation (σx) can be computed using shortcut formulas, which are much simpler than general formulas for discrete random variables but yield equivalent results:
- Mean: $µ_x = np$
- Standard Deviation: $σ_x = \sqrt{np(1 - p)}$
Example: Calculate the mean and standard deviation for the earlier example where n = 8 and p = 0.2.

Partner Up: Critical Thinking Activity

In StatCrunch, navigate to Applets > Sampling Distributions and select Binary with p = 0.2.
Scroll to Statistic(s), select Count instead of Proportion, and perform the calculation. Sample a size of 10, then press 1000 times.
Questions to consider:
- What should the mean be based on the formula?
- What is the mean observed in the simulation?
- What do you notice about the shape of the distribution?
- Repeat the procedure with sample sizes of 20, 50, and 100. Observe any patterns or changes in shape.
Reflect on the significance of these observations and if they correlate with any previous rules or concepts learned.
Execute the same exercise using probabilities p = 0.5 and p = 0.8 and record all observations accordingly.