H2 Mathematics - Binomial Distribution

Title Junior College ‘A’ Levels H1/H2 – Binomial Distribution Author AprilDolphin Date 25/3/2025 Conditions for a random variable to be modelled by a Binomial Distribution includes the following: The experiment must consist of Bernoulli trials (Where there are two possible outcomes in the experiment, which we can call as “outcome” and “complement outcome”.) All trials in the experiment have to be independent (Where the probability of obtaining “outcome” of each trial isn’t affected by a previous trial or will affect a future trial within the experiment). All trials within the experiment have to be identically distributed. (Such that each Bernoulli trial has constant probability of obtaining “outcome” and “complement outcome”.) Example of common outcomes and complement outcomes as follows: Outcome Complement Outcome Yes No No Yes Success Failure Failure Success Picking a red ball Not picking a red ball If the experiment in question satisfies the above requirements, it is said the follow a Binomial Distribution with parameters 𝑛 and 𝑝,where, 𝑛 refers to the total number of trials. 𝑝 refers to the probability of obtaining the “outcome” of each trial. When it is written in standard Binomial Notation, it looks like the following, where 𝑋 refers to the random variable. 𝑋~𝐵(𝑛, 𝑝)

The formula for Binomial Probability distribution of a specific number of trials to be calculated for is given below: 𝑃(𝑋 = 𝑥) = (𝑛 𝑥) 𝑝𝑥(1 − 𝑝)𝑛−𝑥 The formula for Binomial Probability distribution from 0 up till 𝑥 number of trials can be calculated as follows: 𝑃(𝑋 ≤ 𝑥) = (𝑛 0) 𝑝0(1 − 𝑝)𝑛−0 + (𝑛 1) 𝑝1(1 − 𝑝)𝑛−1 + (𝑛 2) 𝑝2(1 − 𝑝)𝑛−2 + ⋯ + (𝑛 𝑥) 𝑝𝑥(1 − 𝑝)𝑛−𝑥 Since A Levels permit the use of Texas Instrument Graphing Calculators in exam condition, I would also have to demonstrate the two rather commonly used functionality in TI-84 Plus CE, namely BinomialPDF and BinomialCDF that is equivalent the above two respectively. BinomialPDF can be used when you are tasked to find 𝑃(𝑋 = 𝑥) given parameters 𝑛 and 𝑝. Example 1. Given the random variable 𝑋~𝐵 (3, 1 6), find 𝑃(𝑋 = 2). This case requires the use of BinomialPDF functionality, which can be accessed by pressing the following buttons on the TI-84 PLUS CE in the following order: Press [2ND] then [VARS] in exact order as mentioned and press the down arrow key repeatedly until you see your calculator cursor reaching an option called “binompdf”. Press [ENTER] key on the calculator and you should see something similar to the below example on the screen trials: p: x value: Paste Key in value of 𝑛 into the number of trials, and press [ENTER] Key in value of p into the field “p” and press [ENTER] Key in number of trials being computed for into the field “x value” and press [ENTER] Once the cursor is on the “Paste”, you should have the following,

trials: 3 p:1/6 x value: 2 Paste Press [ENTER] after checking if the values are co