Binomial Distribution By Dr MAHAMOOD USMAN KHAN Ph D IIT DHANBAD M Sc M Phil AMU Bernoulli Distribution Bernoulli Distribution A Bernoulli experiment is a random experiment the outcome of which can be classified in one of two mutually exclusive and exhaustive ways say success or failure e g female or male life or death nondefective or defective In real life situations in tossing a coin either head or tail will appear in any tossing so tossing of coin can be regarded as Bernoulli trial Similarly while spraying insecticide in a field of crop one can expect that either the insect will be controlled or continue to infest the field The probability mass function of Bernoulli distribution is given by xp 0 where x x 1 p 1 p 0 1 p x 1 0 for otherwise Mean and Variance of Bernoulli Distribution Mean Variance p xE xV pq where q 1 p Binomial Distribution Binomial Distribution It counts the number of successes in n Bernoulli trials It is used to modal the phenomena when there are only two types of possible outcomes success or failure and the process is repeated to a fixed number of times The probability mass function of a binomial distribution is given by xp n 0 qpC x xnx n 2 1 0 x otherwise Mean Variance and Moment Generating function of Binomial Distribution Mean Variance MGF xE xV tM np npq q X pe where nt q 1 p Derivation of Mean and Variance of Binomial Distribution Derivation of Mean and Variance of Binomial Distribution Real life example of Binomial Distribution Real life example of Binomial Distribution in different Fields in different Fields Clinical trials If a new drug is introduced to cure a disease it either cures the disease it s successful or it doesn t cure the disease it s a failure Gambling If you purchase a lottery ticket you re either going to win money or you aren t Medical Number of Side Effects from Medications Banking Number of Fraudulent Transactions Electronic Media Number of Spam Emails per Day Geography Number of River Overflows Business Number of defectives non defectives products Solved Examples Solved Examples Example 1 Suppose it is known that 5 of adults who take a certain medication experience negative side effects Find the probability that more than a certain number of patients in a random sample of 100 will experience negative side effects P X 5 patients experience side effects 0 38400 P X 10 patients experience side effects 0 01147 P X 15 patients experience side effects 0 0004 And so on This gives medical professionals an idea of how likely it is that more than a certain number of patients will experience negative side effects Example 2 Suppose it is known that a given river overflows during 5 of all storms If there are 20 storms in a given year we can use a Binomial Distribution Calculator to find the probability that the river overflows a certain number of times P X 0 overflows 0 35849 P X 1 overflow 0 37735 P X 2 overflows 0 18868 And so on This gives the parks departments an idea of how many times they may need to prepare for overflows throughout the year Example 3 Business Example 3 Business Example 4 Agriculture Example 4 Agriculture Problem It is claimed that 70 insects will die upon spraying a particular insecticide on cauliflower Five insects in a jar were subjected to particular insecticide find out the probability distribution of the number of insect that responded Also find out the probability i that at least three insects will respond and ii more than three insects will respond Solution According to Binomial law of probability the probability of x insects die is No of trials n 5 Probability of success p 70 100 x number of insects die success XPi XPii 3 3 3 0 4 0 XP 3087 0 XP 0 3602 0 5283 4 5 XP XP 0 1681 0 8369 3602 5 XP 1618 Assumptions of Binomial Distribution Assumptions of Binomial Distribution 1 There are a fixed number of trials represented by the variable n 2 Each trial has 2 outcomes called success and failure for convenience 3 The trials are all independent of one another 4 The probability of success is the same for all trials this probability is represented by p and q represents the probability of failure the opposite of success Properties of Binomial Distribution Properties of Binomial Distribution i For a random variable its mean and variance are np and npq respectively As p and q both are fractions generally mean is always greater than variance pnBX ii The variance of binomial distribution is maximum when p 1 2 and the maximum be k independent binomial variates with parameters pni then XX variance is n 4 If kX k k pn i B X 2 1 i i 1 i 1 iii Formula Example Conclusion Thank You Thank You