RI S2B Binomial Distributions_Lecture Notes

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 _____________________________ Chapter S2B: Binomial Distribution Page 1 of 10 Chapter S2B: Binomial Distribution SYLLABUS INCLUDES  Concept of binominal distribution B( , )np as an example of a discrete probability distribution and use of B( , )np as a probability model, including conditions under which the binomial situation is a suitable model  Use of mean and variance of binomial distribution (without proof) PRE-REQUISITES  Probability  Discrete Random Variables CONTENT 1 Binomial Distribution 1.1 Binomial Experiment 1.2 Binomial Random Variable 1.3 Binomial Distribution 1.4 Graph of the Probability Distri bution of a Binomial Random Variable INTRODUCTION Some random variables occur so frequently in real-life that they are given special names. In this chapter, we will look at one such prob ability distribution for di screte random variables, namely, the Binomial distribution. 1 BINOMIAL DISTRIBUTION 1.1 Binomial Experiment Experiments consisting of n trials, each with two possible outcomes that may be regarded as either success or failure, are very common in the study of probability. If, in addition, the probability of getting a success (or a failure) at each trial remains constant and the trials are independent, then we called such experiments binomial experiments. For example, when we toss a coin 10 times, the outcome of each toss may be a head or a tail. Let us regard getting a head as a success. Since we are using the same coin, the probability of success will remain constant. O bviously, the tosses are independent of each other. This coin-tossing experiment is then a binomial experiment.

Raffles Institution H2 Mathematics 2025 Year 6 ___________________________________________________________________________________________ _____________________________ Chapter S2B: Binomial Distribution Page 2 of 10 A binomial experiment is one that possesses the following properties: 1. It consists of n independent trials. 2. The outcome of each trial is either a success or a failure. 3. The probability of success for each trial, denoted by p, remains constant. Now consider drawing 5 cards from a deck of playing cards, one at a time and without replacement. Let us regard the outcome of each draw a success if the card drawn is a King and a failure if it is not. Then such an experiment is not a binomial experiment as we could see that it obviously violates property 1 of binomial experiment. Question: What if the card is replaced before the next draw? Will this experiment be a binomial experiment? Answer: Yes, it will satisfy all properties of a binomial experiment. i.e. (1) It consists of 5 independent trials. (2) The outcome is either a success (if the card drawn is a King) or a failure (if the card drawn is not a King). (3) The probability of drawing a King for each draw r