Binomial Distribution
This is a probability distribution in which the frequencies of happening of exactly, r events in n trials are determined by the model nCrPrqn-r, When the probability of happening of the event in a single trial is known. This model is based on the binomial theorem that was propounded by Jacob Bernoulli, a Swiss mathematician (1654 – 1705) in 1700, and was published fir the first time in 1713, eight years after his death. Hence, this theoretical distribution is widely known after his name as Bernoulli distribution as well.
In the words of Patterson, “A Bernoulli process is a process, wherein an experiment is performed repeatedly yielding either a ‘success’ of a ‘failure’ in each trial, and where there is absolutely no pattern in the occurrence of successes and failures. That is, the occurrence of a ‘success’ or a ‘failure’ in a particular trial does not affect and is not affected by the outcomes in a previous, or subsequent trials. The trials are independent”. nCrPrqn-r, thus cited above has been developed by Bernoulli under under the following assumptions:
(i) The number of trials is repeated for a fixed number of times, i.e., n’ is finite.
(ii) In each trial, there are only two possible outcomes of the event which are mutually exclusive.
(iii) The probability of happening of the event (success) or P remains constant in all the trials.
Similarly, the probability of not happening of the event (failure) or q, also, remains constant in all the trials. For example, the probability of getting a head, or a tail is constant for all the throws of a fair coin. This assumption is very important for that we cannot get a binomial distribution, if the probability of ‘success’ and ‘failure’ do not remain constant in each of the trials. For example, if three cards are drawn, one after another, at random from a pack of playing cards, it will be a binomial distribution, if each card is replaced before another is drawn for that in that case, the probability of drawing a particular type of card will remain the same in all the drawals. On the other hand, if the cards are that in such a case, the probability of drawing a particular type of card will change in each draw.
(iv) All the trials are independent of each other so that the result of any trial is not affected in any way by the result of its preceding trial nor does it affect the result of its succeeding trials. For example, the probability of a ‘head’ on the second tossing of a fair coin is not at all affected by what happened in its first tossing nor such probability affects what would happen in the third tossing of the coin.
