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 remains constant. 1.2 Binomial Random Variable The random variable X denoting the number of successes in the n trials of a binomial experiment is called a binomial random variable. It takes values {0,1, 2,..., }n . The probability that X takes the value x, denoted by P( )Xx , depends on the number of trials (n) as well as the probability of success (p) for each trial. The probability distribution of this discrete random variable is called a binomial distribution. We called n and p the parameters of the distribution and we write B( , )Xn p . Example 1 A coin is biased such that the probability of obtaining a head when the coin is tossed is 2 3 . The coin is tossed 3 times. If X represents the number of heads obtained, find P( ) Xx for 0, 1, 2, 3x and tabulate the probability distribution of X. Solution: 3 11P( 0) P(3 tails) 32 7X     2 3 1 21 2P( 1) P(1 head, 2 tails) C 33 9X        2 3 2 214P( 2) P(2 heads, 1 tail) C 339X       

Raffles Institution H2 Mathematics 2025 Year 6 ___________________________________________________________________________________________ _____________________________ Chapter S2B: Binomial Distribution Page 3 of 10 3 28P( 3) P(3 heads) 32 7X     x 0 1 2 3 P( )Xx 1 27 2 9 4 9 8 27 Note: The random variable X representing the number of heads obtained is a binomial random variable. The number of trials (n) is 3 while the probability of success (p), i.e. the probability of obtaining a head, for each trial is 2 3 . We write 2B3 , 3X    . In general, if B( , )Xn p , how do we evaluate P( )Xx for 0,1, 2,...,x n ? We seek a formula that expresses P( )Xx in terms of x, n and p. 1.3 Binomial Distribution If B( , )Xn p where X denotes the number of successes in n trials of a binomial experiment, then the probability distribution of X is given by P( ) (1 ) x nxnXx p p x    , for 0,1, 2,...,x n , where ! (! ! n x n nCx nx ) x    This is called the binomial distribution with parameters n and p. The mean and variance of X are given by (i) E( )Xn p (ii) Var( ) (1 )Xn p p [Highlighted formulae can be found in formula list] Note: When n = 1, the binomial distribution is known as the Bernoulli distribution. Example 2 An ordinary fair die is tossed eight times. Find the probability of obtaining at least 6 sixes. Solution: Let X be the number of sixes obtained in 8 tosses of the die. Then 1B8 , 6X    . Probability of obtaining at least 6 sixes = P( 6)X  P( 6, 7, 8)X 62 7 8 88 67 15 15 1 0.00044166 66 6CC            (3 s.f.)

Raffles Institution H2 Mathematics 2025 Year 6 ___________________________________________________________________________________________ _____________________________ Chapter S2B: Binomial Distribution Page 4 of 10 Example 3 A binomial random variable X has mean 1.2 and variance 1.08. Evaluate the parameters of the distribution. Solution: B( , )Xn p . We have E( ) 1.2X  and Var( ) 1.08X  . 1.2np and (1 ) 1.08np p (1.2)(1 ) 1.08p  1 0.9 0.1pp  and 12n  B1 2 , 0 . 1X  Note: For special discrete random variables such as binomial distribution, stating B( , )Xn p is equivalent to listing all probabilities of the probability distribution. 1.4 Graph of the Probability Distribution of a Binomial Random Variable Suppose B( , )Xn p . The graphs of the probability distribution of X for various values of n and p are shown below. In general, as x increases, P( )Xx will increase to a maximum value and after which, it will decrease. 10 20 30 0.1 0.2 0.3 0.4 x P(X=x) n = 25, p = 0.2 10 20 30 0.1 0.2 0.3 0.4 x P(X=x) n = 25, p = 0.9 10 20 30 0.1 0.2 0.3 0.4 x P(X=x) n = 25, p = 0.5 2 4 6 0.1 0.2 0.3 0.4 0.5 0.6 x P(X=x) n = 5, p = 0.2 2 4 6 0.1 0.2 0.3 0.4 0.5 0.6 x P(X=x) n = 5, p = 0.5 2 4 6 0.1 0.2 0.3 0.4 0.5 0.6 x P(X=x) n = 5, p = 0.9

Raffles Institution H2 Mathematics 2025 Year 6 ___________________________________________________________________________________________ _____________________________ Chapter S2B: Binomial Distribution Page 5 of 10 Example 4 (a) The probability that a sharp shooter hits a ta rget is 0.8 and the probability that he misses is 0.2. The shots he makes are independent of each other and the probability of him hitting or missing the target remains constant. Find th e probability that, in 10 shots, he will hit the target (i) exactly 6 times, (ii) more than 8 times. Find also, in 10 shots, the most likely number of shots that he will hit target. (b) A poor shooter has a probability of 0.1 of h itting the target. How many shots must he be given in order that there is at least a 90% chance that he will hit the target at least