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 countable number of values. 3. Let Y be the random variable denoting the number of accidents reported on an expressway in 4 hours. Y is a discrete random variable because it is possible to have infinite number of accidents in 4 hours but the values are countable. Note: We usually denote a random variable by a capital letter e.g. X, Y, T, …, the particular value it takes by a lower case letter e.g. x, y, t, …

SAJC 2021 JC 1 H1 Mathematics Chapter 3: Binomial Distribution Binomial Distribution Page 3 of 31 Discuss Which of the following is a discrete random variable and why? 1. 4 cards are drawn from a pack of playing cards. Let X be the random variable “the number of jacks obtained when 4 cards are drawn from a pack of playing cards ”. Yes. X can take values x = 0,1,2,3,4. (countable number of values) 2. Let Y be the random variable “The height of a randomly chosen 18-year-old student” No. Y can any values in a given range. 3.2 Motivation In the previous chapter, we learnt about calculating the probability of an event occurring in a given experiment. Let us consider the following: Scenario: A biased coin is being tossed 3 time s. The probability of getting heads in each toss is 2 3 . Calculate the probability of getting: (i) no heads (ii) exactly two heads. Method: Since the probability of getting heads for each toss of the coin is known to be 2 3 , therefore the probability of getting tails for each toss of the coin is 1 3 . Let us then draw the probability tree for the experiment where the coin was tossed 3 times:

SAJC 2021 JC 1 H1 Mathematics Chapter 3: Binomial Distribution Binomial Distribution Page 4 of 31 Using this probability tree, we could then calculate that: (i) P(no heads was obtained) = P(TTT) = 3 11 3 27   (ii) P(Exactly two heads were obtained)       22 P HHT P HTH P THH 2 1 2 1 2 1 2 3 3 3 3 3 3 3 4 9                                      Note that in both events, we observed the possible outcomes that would gi ve us no heads or exactly two heads ({TTT} and {HHT, HTH, THH} respectively, and calculated the probability of each outcome separately. H T H T H T H T H T H T H T 2 3 1 3 1st toss 2nd toss 3rd toss 1 3 1 3 1/3 1/3 1/3 1/3 2 3 2 3 2/3 2/3 2/3 2/3

SAJC 2021 JC 1 H1 Mathematics Chapter 3: Binomial Distribution Binomial Distribution Page 5 of 31 Example 1 Based on the example in Section 3.2, what is the probability of getting (i) no heads, (ii) exactly one head, (iii) exactly two heads, (iv) all heads. Solution: Let X be the random variable denoting the number of heads obtained when the biased coin is tossed 3 times. P( )Xx Calculation (i) P( 0)X    3 3 11P TTT 0 3 27    (ii) P( 1)X        2 3 2 1 2P HTT P THT P TTH 1 3 3 9              (iii) P( 2)X        2 3 2 1 4P HHT P HTH P THH 2 3 3 9                 (iii) P( 3)X    3 3 28P HHH 3 3 27    We can therefore deduce that 3 3 21P( ) where 0,1, 2,333 xx X x x x                . For a scenario with 3 coin flips, it is still easy to manually calculate the probability of 0, 1, 2, or 3 heads. However, what would happen if we were asked to consider a scenario with 100 coin flips? 1000 coin flips? Such calculations would be too tedious. This is where the generalisation above comes in (for  P Xx ). Using the above expression, we can easily calculate the probability of x heads, given that we know the number of coin flips and the probability of obtaining heads. This idea of generalising for experiments with only two outcomes is the motivation behind this chapter, Binomial Distribution.

SAJC 2021 JC 1 H1 Mathematics Chapter 3: Binomial Distribution Binomial Distribution Page 6 of 31 3.3 Binomial Distribution Definition Let X be the random variable denoting the number of successes in n trials of a bi nomial experiment, then X is said to follow a Binomial distribution. This is written as: X  B(n, p) Note that the number of trials , n, and the probability of success of a trial, p, are required to describe the binomial distribution. The values of n and p are known as the parameters of the distribution. If X  B(n, p), the probability of obtaining exactly x successes in n trials is denoted by P(X = x), where P( ) C n x n x xX x p q  for x = 0, 1, 2, …, n . (given in MF26) (p is the probability of success and q = 1 – p is the probability of failure) Note: 1. 00 0P( 0) C n n nX p q q    2. P( ) C n n n n n nX n p q p    3. (q + p)n = 0 1 2 2 0 12 ...n n n n n n n n onC p q C pq C p q C p q    = P(X = 0) + P(X