SAJC 2021 Binomial Distribution (Teacher)

SAJC 2021 JC 1 H1 Mathematics Chapter 3: Binomial Distribution Binomial Distribution Page 1 of 31 Chapter 3 (Statistics): Binomial Distribution (Teacher’s copy) Objectives At the end of the chapter, you should be able to (a) understand the concepts of discrete random variable and binomial variable; (b) understand that the binomial distribution B( n, p) is a probability distribution with mean np and variance  1np p ;  know the conditions under which the binomial distribution is a suitable model and able to comment on the appropriate use of the model and the assumptions made. (c) understand th e relationship between binomial probabilities and the binomial expansion of   n ab for positive integer n;  use of the notations !n and n r   (d) use of graphic calculator to calculate probabilities; (e) use the binomial distributions to model practical situations. Content 3.1 Random Variable 3.1.1 Discrete Random Variable 3.2 Motivation 3.3 Binomial Distribution 3.4 Conditions on Binomial Distribution 3.5 Use of Graphic Calculator for Binomial Distributions 3.6 Mean and Variance of a Binomial Distribution 3.7 Mode of a Binomial Distribution

SAJC 2021 JC 1 H1 Mathematics Chapter 3: Binomial Distribution Binomial Distribution Page 2 of 31 3.1 Random Variable From the previous chapter, we learnt to calculate probabilities for different outcomes of an experiment. A random variable is a qualitative variable that takes different numerical values according to the results of the experiment. Random variables are represented by capital letters X, Y, … and the possible observed values of X, Y, ... are represented by lower case letters x, y, … .[avoid using Z, B, N, P and T] Consider the experiment when a fair coin is tossed twice. The sample space,  S HH, HT, TH, TT We can let X represents the number of heads obtained when a fair coin is tossed twice. We observe that  the possible values of X are 0, 1, 2 (Numerical Values)  the value it obtained is subjected to chance (Random) Hence, X is a random variable. The expression 1X  describes the event of obtaining 1 head when a fair coin is tossed twice. The value of  P1X  is the probability of obtaining 1 head when a fair coin is tossed twice. Therefore  P1X  = 1 1 1 1 1P(H,T) P(T,H) 2 2 2 2 2      3.1.1 Discrete Random Variable A discrete random variable is one which can only take a finite or countably infinite number of values. Examples of discrete random variables: 1. A fair coin is tossed 4 times. Let X be the random variable denoting the number of heads obtained. Then X is a discrete random variable as X can only take exact values x = 0, 1, 2, 3, 4. 2. Let X be the random variable denoting the number of eggs that a hen lays in a day. This is a discrete random variable because its only possible values are x = 0, 1, 2,..., i.e. there is a co