Maharashtra Board Class 12 Maths Part 2 Chapter 8 Binomial Distribution PDF Download

Binomial Distribution

Let us Study

Bernoulli Trial

Binomial distribution

Mean and variance of Binomial Distribution.

Let us Recall

Many experiments are dichotomous in nature. For example, a tossed coin shows a head or tail. A result of student pass or fail. A manufactured item can be defective or non-defective. The response to a question might be yes or no. An egg has hatched or not hatched. The decision is yes or no. In such cases, it is customary to call one of the outcomes a success and the other not success or failure. For example, in tossing a coin, if the occurrence of the head is considered a success, then occurrence of tail is a failure.

Let us Learn

Bernoulli Trial

Each time we toss a coin or roll a die or perform any other experiment, we call it a trial. If a coin is tossed, say, 4 times, the number of trials is 4. Each trial has exactly two outcomes, namely, success or failure. The outcome of any trial is independent of the outcome of any other trial. In each of such trials, the probability of success or failure remains constant. Such independent trials which have only two outcomes usually referred to as success or failure are called Bernoulli trials.

Definition

Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions:

Each trial has exactly two outcomes: success or failure.

The probability of success remains the same in each trial.

Throwing a die 50 times is a case of 50 Bernoulli trials. In which each trial results in success (say an even number) or failure (an odd number). The probability of success p is same for all 50 throws. Obviously, the successive throws of the die are independent trials. If the die is fair and has six numbers 1 to 6 written on six faces, then \(p = \frac{1}{2}\) and \(q = 1 - p\), so \(q = \frac{1}{2}\)

Teacher's Note

Bernoulli trials are like tossing a coin many times. Each toss has only two results: heads or tails. Just like when you take a test at school, each question has success (correct answer) or failure (wrong answer).

Exam Trick

Remember: Bernoulli trial = two outcomes only = success or failure. Write this formula in your memory: p + q = 1. If p is success, then q is always failure.

Points to Remember

Bernoulli trial has only two outcomes.

Probability of success stays the same every time.

Each trial is independent of other trials.

We use p for success and q for failure.

For example:

Consider a die to be thrown 20 times. If the result is an even number, consider it a success, else it is a failure. Then \(p = \frac{1}{2}\) as there are 3 even numbers in the possible outcomes.

If in the same experiment, we consider the result a success if it is a multiple of 3, then \(p = \frac{1}{3}\) as there are 2 multiples of 3 among the six possible outcomes. Both above trials are Bernoulli trials.

Solved Example

Ex. 1: Six balls are drawn successively from an urn containing 7 red and 9 black balls. Tell whether or not the trials of drawing balls are Bernoulli trials when after each draw the ball drawn is (i) replaced (ii) not replaced in the urn.

Solution:

(i) The number of trials is finite. When the drawing is done with replacement, the probability of success (say, red ball) is \(p = \frac{7}{16}\) which is same for all six trials (draws). Hence, the drawing of balls with replacements are Bernoulli trials.

(ii) When the drawing is done without replacement, the probability of success (i.e. red ball) in first trial is \(\frac{7}{16}\), in second trial is \(\frac{6}{15}\) if first ball drawn is red, and is \(\frac{7}{15}\) if first ball drawn is black. And so on. Clearly probability of success is not same for all trials, hence the trials are not Bernoulli trials.

Teacher's Note

When you draw a ball from a bag and put it back, this is Bernoulli trial. But if you do not put it back, it is not Bernoulli trial. Like in India when you draw a raffle ticket, if you put it back, the chance stays the same.

Exam Trick

Remember: With replacement = Bernoulli trial. Without replacement = NOT Bernoulli trial. This is because when you replace, the probability stays same.

Points to Remember

Replacement makes probability stay the same.

Without replacement changes probability every time.

Always check if probability is constant.

Binomial Distribution

Consider the experiment of tossing a coin in which each trial results in success (say, heads) or failure (tails). Let S and F denote respectively success and failure in each trial. Suppose we are interested in finding the ways in which we have one success in six trials. Clearly six different cases are there as listed below:

SFFFFF, FSFFFF, FFSFFF, FFFSFF, FFFFSF, FFFFFS.

Similarly, two successes and four failures can have \(\frac{6!}{4! \times 2!} = 15\) combinations. But as n grows large, the calculation can be lengthy. To avoid this the number for certain probabilities can be obtained with Bernoulli's formula. For this purpose, let us take the experiment made up of three Bernoulli trials with probabilities p and q = 1 – p for success and failure respectively in each trial. The sample space of the experiment is the set

\(S = \{SSS, SSF, SFS, FSS, SFF, FSF, FFS, FFF\}\)

Teacher's Note

Binomial distribution helps us find the probability of success when we repeat an experiment many times. In India, if a shop receives 100 bulbs and 5% are defective, we use binomial distribution to find how many defective bulbs we might get.

Exam Trick

Remember: Binomial = two outcomes. The word "bi" means two. Just like a bicycle has two wheels, binomial has two outcomes: success and failure.

Points to Remember

Binomial distribution uses Bernoulli trials.

We can count successes in many trials.

The formula uses combination: nCx.

Probabilities come from binomial expansion.

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