Binomial formula deriviation

[Anon]

Can anybody write down deriviation of binomial formula. It is'nt in my course book. Looking forward

 

[PlanetMaster]

When you have a binomial distribution, these criteria are met:
- There is a set number of trials, n
- There are only two possible outcomes from each trial, a success and a failure
- The probability of success, p, is fixed
- Each trial is independent of the other trials

If we have only two outcomes then:
p(success) = p
p(failure) = 1 - p

We want the probability of getting "k" successes. That means we must be successful k times and fail n - k times. So lets say k=3, n=5, and p=0.7, that means:
P(success) AND P(success) AND P(success) AND P(failure) AND P(failure)
AND keywords in probability mean multiply:
P(success) * P(success) * P(success) * [1-P(success)] * [1-P(success)]
0.7 * 0.7 * 0.7 * 0.3 * 0.3
(0.7)^3 * (0.3)^2

So in a general form we can say the probability of getting k successes given n trials where the probability of success of each trial is p is:
P(x = k) = p^k * (1-p)^(n-k)

But this isn't complete yet. We kind of made it inferred that we need to get 3 successes in a row followed by 2 failures. But this is not the case. This means that we need to "alter" this probability to allow for varying alignments of successes/failures. This means doing "n choose k" to allow for varying permutations. "n choose k" (which is n! / (k!(n-k)!)) finds how many different ways you can "k" things from "n" total things. So for instance in our scenario:
SSSFF
SSFSF
SFSSF
FSSSF
SSFFS
SFFSS
FFSSS
...etc
however many different ways I can write this combination are accounted for by this formula.
Multiply this what we found before and it's done.

P(x=k) = (n choose k)(p^k)((1-p)^(n-k))

 

[Anon]

Thank you very much specially on such a speedy reply.

 

[NoMI]

which subjct?

 

[anzaar]

the above derivation relates to Binomial Distribution(Statistics).
here is the other way

 

[anzaar]

the above derivation relates to Binomial distribution(statistics) for ALevel.
here is other way of explanation.

The Binomial Theorem

A binomial is an algebraic expression containing 2 terms. For example, (x + y) is a binomial.
We sometimes need to expand binomials as follows:
(a + b)^0 = 1 Power is 0 so one term
(a + b)^1 = a + b Power is 1 so two terms
(a + b)^2 = a^2 + 2ab + b^2 Power is 2 so three terms
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4
(a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5
Clearly, doing this by direct multiplication gets quite tedious and can be rather difficult for larger powers or more complicated expressions.
Pascal's Triangle
We note that the coefficients (the numbers in front of each term) follow a pattern.
[This was noticed long before Pascal, by the Chinese.]
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
You can use this pattern to form the coefficients, rather than multiply everything out as we did above.
The Binomial Theorem
We use the binomial theorem to help us expand binomials to any given power without direct multiplication. As we have seen, multiplication can be time-consuming or even not possible in some cases.
Properties of the Binomial Expansion (a + b)^n ;if power is n so n+1 terms
• There are n + 1 terms.
• The first term is a^n and the final term is b^n.
• Progressing from the first term to the last, the exponent of a decreases by 1 from term to term while the exponent of b increases by 1. In addition, the sum of the exponents of a and b in each term is n.
• If the coefficient of each term is multiplied by the exponent of a in that term, and the product is divided by the number of that term, we obtain the coefficient of the next term.
General formula for (a + b)^n
First, we need the following definition:
Definition: n! represents the product of the first n positive integers i.e.
n! = n(n − 1)(n − 2) ... (3)(2)(1)
We say n! as 'n factorial'
Examples:
3! = (3)(2)(1) = 6
5! = (5)(4)(3)(2)(1) = 120
4!/2!={(4)(3)(2)(1)} /{(2)(1)}=12

Note : cannot be cancelled down to 2!
Binomial Theorem Formula
Based on the binomial properties, the binomial theorem states that the following binomial formula is valid for all positive integer values of n:
(a+b)^n=(a)^n + n(a)^(n-1)(b) + {n(n-1)(a)^(n-2)(b)^2}/2! +{n(n-1)(n-2)(a)^(n-3)(b)^3}/3!+......(b)^n
This can be written more simply as:
nC0(a)^n + nC1(a)^(n-1)(b) + nC2(a)^(n-2)(b)^2 + nC3(a)^(n-3)(b)^3 + ......(b)^n
We can use the nCr button on our calculator to find these values.
________________________________________
These are usually written nCr.
Binomial Series
Note: here some typical expression cant be written . for more information you may email
at [email protected]. i can send whole notes on Binomial.
regards
anzaar.

 

[Anon]

@anzaar Thank you very much. Your explanation was quite explicit.

 

[NoMI]

mere to uper se guzar gai

 

[anzaar]

@anon You are always welcome.
for any query, any problem just post. i will try to reply specifically.

 

[NoMI]

hmm