Chapter 8 Binomial Distribution

Additional Mathematics Form 5

1

Chapter 8: PROBABILITY DISTRIBUTION 8.1

Name:……………………………….

The Binomial Distribution

In statistics the so-called binomial distribution describes the possible number of times that a particular event will occur in a sequence of observations. The event is coded binary, it may or may not occur.

(a)

DISCRETE RANDOM VARIABLES

A random variables that has finite and countable values is known as a discrete random variable. For example, two coins are tossed simultaneously and the number of heads obtained is studied. If X represents the number of heads, then X, can take the values of 0 (no head obtains), 1 (1 head obtains) and 2 (2 heads obtains), based on the following tables. From the experiment (Bernoulli’s experiment), the sample, S = {HH , HT , TH , TT }

Outcomes X

HH 2

HT 1

TH 1

TT 0

Hence, X is a discrete random variable. (b)

PROBABILITY OF AN EVENT THAT FOLLOWS A BINOMIAL DISTRIBUTION

The random variable X of a binomial distribution counts the number of successes in n trials. The probability that X is a certain value r is given by the formula P ( X = r )= n C r p r q n −r where r = 0, 1, 2, 3, 4,……, n q = probability of successes in each trial q = probability of failure in each trial

Note: p + q = 1 and q = 1 − p In general, the binomial distribution describes the behavior of a count variable X if the following conditions apply:

1: 2: 3: 4:

The number of observations n is fixed. Each observation is independent. Each observation represents one of two outcomes (success or failure) The probability of success, p is the same for each outcome.

If these conditions are met, then X has a binomial distribution with parameters n and p, abbreviated B (n, p ) . Sri Bintang Tuition Centre 2009, Kuching

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Additional Mathematics Form 5

2

Examples 1.

List all the possible values of a discrete random variable for the followings: (a) Tossing a coin 3 times with X represents the number of tails obtained. (b) A box contains 3 blue marbles and 2 white marbles. Two marbles are picked at random from the box. X represents the number of blue marbles drawn. (c) In a shooting training, a man is given 6 trials. X represents the number of trials that hit the target.

2.

P( X = x) 2y

y

0

1

2

3

x

The diagram shows the graph of binomial distribution for X. Find the value of y.

3.

A random variable X has a binomial distribution. Given the number of trials and the probability of success are 8 and 0.48 respectively. Find P ( X = 5) .

4.

A fair coin is tossed 5 times one after another. Find the probability of getting a “heads” twice.

Sri Bintang Tuition Centre 2009, Kuching

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Additional Mathematics Form 5

3

5.

A box contains 3 red cards and 4 blue cards. A card is taken at random from the box with replacement. Find the probability that a blue card is chosen 4 times out of 6 times.

6.

The probability that a school basketball team will win in the inter-school games is 0.65. Find the probability that the team will win (a) exactly 4 times, (b) at least 5 times.

7.

In the training of penalty kicks for a soccer, the chance for Amer to score the goal is 0.85. Amer tries 10 kicks. Find the probability that (a) Amer scored 8 goals, (b) at least 2 goals, (c) at most 3 goals.

Sri Bintang Tuition Centre 2009, Kuching

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Additional Mathematics Form 5

4

8.

In a Form 5 class, it was found that 60% of the students have a mobile phone. If 4 students are chosen at random, determine the binomial distribution for the number of students who have a mobile phone. Sketch the graph of this binomial distribution.

9.

85% of the bulbs produced by a factory are accepted by SIRIM. If 10 bulbs are chosen at random, find the probability that (a) at least nine are accepted by SIRIM, (b) none is rejected.

10.

In a certain class, 40 out of 60 students have a personal computer at home. If 12 students are chosen at random from the group, find the probability that (a) 4 of them have a personal computer, (b) 2 or more have a personal computer.

Sri Bintang Tuition Centre 2009, Kuching

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Additional Mathematics Form 5

5

Mean, Variance & Standard Deviation of Binomial Distribution

• • •

Mean, µ = np Variance, σ 2 = npq Standard deviation, σ = npq

11.

A test has 40 multiple-choice questions with options, A, B, C, D and E. Given that there is only one right answer for each question and answers to each question are done by guessing, find (a) the mean of the right answers, (b) the variance and the standard deviation of the number of right answers.

12.

A group of letters written on each card is placed in a box. Each card is drawn from the box with replacement before the next card is picked. The process is repeated n times. Given that the mean of getting a letter “A” is 120 and the variance is 40. Find the value of n.

13.

The probability that an archer hits the target is p. After a training, it was found that the mean score was 4.8 when the archer was given 8 chances. (a) Find the value of p. (b) If an archer is chosen at random, find the probability that he will hit the target at least once.

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