You may find these examples helpful:

http://onlinestatbook.com/2/probability/binomial.html

or

https://people.richland.edu/james/lecture/m170/ch06-bin.html

Danke Schon :)

It is only success vs failure or happening vs non-happening of the event. The probability of success should be remain same. Success of crop or plantation at farmers field with same species and same management condition.

When you make some observations where two possible results are possible, than you observe a "binomial variable" (bi-nomial: the values of the variable can take two possible different "names"). Examples:

toss of a coin (head or tail)

result of an exam (pass or fail)

sex of offspring (male or female)

therapeutic effect (cure or effectless)

5-year survival after diagnosis of a disease (alive or dead)

existence of a particular mutation (yes or no)

...

These all are binomial variables, because the values can always be one of two different possibilities. In general, the two possible outcomes are denoted as "success" and "failure" (what you consider as "success" is your chioce - it does not matter anyway; these are just two names).

If you now make a *series* of n such observations (n is some natural number), you get k (also some number smaller or equal to n) "successes" and (n-k) "failures".

The binomial distribution allows you to calculate the expectation to observe 1 or 2 or 3 or... k "successes" in a series of observations, based on the expectation to observe a "success" in one particular observation (often indicated with the symbol p or π (pi).

Example: If you toss a coin you may expect "head" as being equally likely as "tail". In a series of 10 tosses, how likely would you then expect less than three times "head" (and more than 7 times "tail")? This can be calculated using the binomial distribution.

One sais: the number of "heads" (k) in n coin tosses with p = P(head) in a single toss is binomial distributed with B(k | n,p).

The probability to get less than three times head is P(0 x head or 1x head or 2 x head) = P(0 x head) + P(1 x head) + P(2 x head)

P(0 x head) = B(k=0 | n=10, p=0.5) = 0.098%

P(1 x head) = B(k=1 | n=10, p=0.5) = 0.98%

P(2 x head) = B(k=2 | n=10, p=0.5) = 4.39%

The sum is 0.098+0.98+4.39 = 5.5%

When usually 30% of the students fail in an exam (so p=0.3), the number k of students that will fail in a class of n=25 students is binomial distributed with B(k | n=25, p=0.3). From this we may calculate that the probability that all students will pass is B(k=0 | n=25, p=0.3) = 0.13%. The probability that all students will fail is very tiny (B(k=25 | n=25; p=0.3) = (8.5E-12)%). The probability that no more than 5 students will fail is sum[i=0..4](B(k=i | n=25; p=0.3) = 19.3%.

You can exand the other examples above in the same way.

Thanks a lot. It was really helping :)

Bitteschön :)

typical way to introduce binomial distribution by Jochen Wilhelm.

A small correction the example of toss a coin etc. are Bernoulli variables not Binomial variables. ! Binomial is the sum of n independent Bernoulli.

@Subrata: Thanks, you are right: a SINGLE coin toss is a Bernoulli variable. The domain of a Bernoulli variable is binomial or binary or dichotomous, i.e. it can take one of two values. It is usually numerically coded as 1 ("success") and 0 ("failure"). Then, the sum of n such Bernoulli variables is a Binomial variable. The domain of a Binomial variable are the natural numbers (0,1,2,...n).

However, note that for n=1 the Bernoulli variable and the binomial variable are identical. So the Bernoulli variable is just a special case of a binomial variable (one can say that any Bernoulli variable is a binomial variable, but not any binomial variable needs to be a Bernoulli variable). I did not want to introduce so many different names, therefore I deliberately only explained binomial variables.

@Jochen Wilhelm: thanks

This may useful to u Anjani

Binomial distribution was introduced by James Bernoulli in 1700 and published in 1713.

The binomial experiment (distribution) arises in following situation:

(i) the underlying experiment consists of n independent and identical trials;

(ii) each trial results in one of two possible outcomes, a success or a failure;

(iii) the probability of a success in a single trial is equal to p and remains the same throughout the experiment; and

(iv) the experimenter is interested in the r.v. X that counts the number of successes observed in n trials.

Dear Anjany,

I have the perfect example for you! It is the Roman Spring. As it is shown in the picture. Sorry about the Greek. Δοκιμή means trial. Now each molecule of water that comes out, essentially is a binomial trial: Should it go from the left or the right? Suppose you flip a coin and decide analogously. The mass of water that you see, that comes out of a bowl is almost the limit probability when we have practically infinite number of trials. If you mark each effluence by H(eads) ot T(ail), then you can "see" without calculation, the probability of having e.g. HTTH etc.

There is a paper, unfortunately in Greek, where there are methods od teaching probability in the secondary schools. Have a look, you can see a lot og things just by looking at figures:

https://www.researchgate.net/publication/243962234_____________?ev=prf_pub

Data ΓΙΑ ΤΗΝ ΔΙΔΑΣΚΑΛΙΑ ΤΩΝ ΒΑΣΙΚΩΝ ΕΝΝΟΙΩΝ ΤΗΣ ΠΙΘΑΝΟΤΗΤΑΣ ΣΤΗ Μ...

@Costas Drossos: Sir figures are also difficult to understand it would have been great if it was in English. I found the water example exciting and easy to understand.

I will give you an analytical model:

Binomial:How many "successes" out of a given number of independent trials with same probability of "success".

one example of many ways the binomial distribution is used. Another obvious one is in test scores. The "percentile" you are in after taking an SAT test, for example, is the percent chance that a person taking the test would score lower than you did. The range of possibilities from a person filling in their name then turning in a blank test all they way up to scoring 1 less than you are added up and given as your percentile. They use the actual results to give you your percentile, but the actual results in test scores fit very closely with a mathematical binomial distribution (the bell curve -- low probability of very low scores, low probability of very high scores, higher chances of middle-of-the-road scores).

Many instances of binomial distributions can be found in real life. For example, 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). If you purchase a lottery ticket, you’re either going to win money, or you aren’t. Basically, anything you can think of that can only be a success or a failure can be represented by a binomial distribution.