The distribution of data here is discrete which describes whether a given event has failed or succeeded. It deals with only two possible outcomes. It is different from Normal distribution in the nature of distribution where the latter is continuous.It is used to obtain the probability of observing x successes in N trails. But when we deal with a large dataset even binomial distribution shows similar characteristics to Normal distribution.
Criteria for the Distribution:
The number of trials or observations are fixed: For example, tossing a coin n number of times. This n is a finite number. From this data only we can find the probability of any event example, finding out the probability of head occurring x number of times, where x<=n
Each observation or trial is independent: Taking the above example, if we toss a coin 6 times so the result of the tossed coin will not be impacted by the previous toss or the current result won't impact the next toss.
Each observation represents one of two outcomes("success or failure")
The probability of success is exactly the same from one trial to another.
Mean and Variance of the Binomial Distribution
Steps to Approach a Binomial problem
80% of people who purchase pet insurance are women. If 9 pet insurance owners are randomly selected, find the probability that exactly 6 are women.?
Step1: Identify 'n' from the given problem. Here in this case n=9.(number of randomly selected items).
Step2: Identify 'x' from the problem, whose probability we need to find. Here x=6.
Step3: 1st part of formula: n!/((n-x)!*x!) = 9!/((9-6)!*6!) = 84
Step4: We try to find out the probability of success p and probability of failure (1-p)=q. From here we get p = 0.8 and q= 0.2
Step5: 2nd part of formula: p^x = (0.8)^6 = 0.2621
Step6: 3rd part of formula : q^(n-x) = (0.2)^(9-6) = 0.008
Step7: Multiply the results obtained from steps 3,5 and 6. We get 84*0.2621*0.008 = 0.176