Math Guide

Bernoulli Numbers Problem

Introduction of Bernoulli trails:


Bernoulli trails is the important topic in the probability chapter in mathematics subject. So many experiments are dichotomous in nature. For example, If one coin is tossed means, it shows whether a ‘head’ or ‘tail’, same like that a manufactured item is to be ‘defective’ otherwise ‘non-defective’, the response of the question is might be ‘yes’ or ‘no’ and the decision is ‘yes’ or ‘no’ etc. In some times, it is customary to call one of the outputs were a ‘success’ and some other cases it is ‘not success’ or ‘failure’.

Example: In tossing a coin, if the occurrence of the head is considered a success, then occurrence of tail is a failure. This is called Bernoulli trails.


Bernoulli Numbers Problem:

                Each time we roll a die or toss a coin or perform any other experiment,we call it a trial. If a dies is rolled, say, 5 times, the number of trials is 5, each has two outcomes must, named positive or negative (success or failure). The output of any trial is independent of the output of any other trial. Such a independent trials which are having only two outputs usually referred as ‘successes’ or ‘failure’. Here now we are going to solve the numbers problem using Bernoulli trails.


Conditions:

                    The random experiments which are called Bernoulli trials, if they satisfy the following four conditions: there are given below:

1) It should be a finite numbers of trials only.

2) All the trials should be in independent only.

3) All trials having exactly two outputs: Positivenumbers or negative numbers. (Success or failure)

4) The probability of success remains the same in all trial.


Example Problem for Bernoulli Numbers Problem:


Example problem: 6 balls are drawn successively from a bag 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 bag.


Solution

i)                    Replaced,

                      Here the numbers of trials is finite.

                      When the drawing is done with replacement of balls

                      The probability of success (red ball) is p = `7/16.`

                      This is same for all 6 trials.

 Hence, the drawings of balls with replacements are Bernoulli trials.

ii)                   not replaced

                      When the drawing is done, the probability of success is without replacement. (i.e., red ball) in first trial is 7 /16,

                      In 2nd trial are 6/ 15.

                      If the first ball drawn is red or `7/15.`

                      If the first ball drawn is black and so on.

                      Then, the probability of success is not same as all the trials;

Hence the trials are not Bernoulli trials in this problem.