The expansion is of the type (x + y )n where x, y are two real numbers & n is an integer.
We have (x + y )n = nC0xn y0 + nC1 xn-1y1 + nC2 xn-2y2 + nC3 xn-3y3 +.nC4 xn-4y4 +...........+ nCn-1 x1yn-1 + nCn x0yn
Or (x + y )n = Σ k=0 --> n nCk xn-k yk where nCk is also written as . These are called binomial coefficients and are positive numbers.
1. There are n+1 terms in the expansion. If n is even , no. of terms is odd & vice versa. In case of even terms, all coefficients occur in pairs. In case of odd terms , only the middle coefficient does not occur in pair.
2. Arranging co-efficients in rows for successive values of k from k=0 to k =n gives rise to Pascal's triangle .
3. nCk = n! / [(k!)*(n-k)!] . Therefore nC0 = 1 & nCn = 1 & 0Ck = 0 for all k > 0 .
4. nCk = n-1Ck -1 + n-1Ck for n , k >0
5. nCk =∏ i=1 --> k [n - (k-i)]/i
7) workout on factorials 4
8) binomial co-efficient chart
9) binomial calculation clickCombinations:
The general formula is:
Probability distribution of a random discrete variable is called probability mass function whereas probability distribution of a random continuous variable is called a probability density function.
If k is a random discrete variable and there are n number of trials, and there are only 2 possible outcomes- success (probability-p) & failure (probability-q=1-p), then probability of getting k no. of successes in n number of trials is as per Binomial Distribution --
P(p)= nCk pk
B(k)= nCk pk qn-k where B(k) is called Binomial frequency function of p. ΣB(k)= (p+q)n for all values of k =0 to n. In other words, the
sum of the values of frequency at point k=0,1,2,.....n is equal to (p+q)n
B(k) / B(k-1) =((n+1-k)/k)*p/q =Q
Qk > 1 when (n+1)p > k(p+q) i.e frequency function value at that k is greater than its previous k value.
Qk = 1 when (n+1)p = k(p+q) i.e frequency function value at that k is equal to its previous k value.
Qk < 1 when (n+1)p < k(p+q) i.e frequency function value at that k is less than its previous k value.
Most Probable Frequency : Supposing k has values ni ( n0,n1,n2,.......nn
, )then if for any ni there is a nj
such that B( nj) ≥B( ni) where i,j
=0,1,2,3.....n, then nj is called the most probable value of
k, corresponding frequency B(nj)
being most probable frequency.
Uniform Sample Space: If we toss 2 coins, the sample points are (head ,head),(head,tail),(tail,head),(tail, tail). Each of the 4 sample points have the same probability i.e 1/4. Hence sample space is uniform. A sample space is said to be uniform if each point has the same probability.
Non-Uniform Sample Space: A sample space is said to be non-uniform if each point in it does not have the same probability. For example, no. of heads in tossing of 4 coins. These are nC0 , nC1 ,nC2 , nC3, nC4, which represent probabilities for no head, 1 head, 2 head, 3 head, 4 head respectively and have different values. Hence, the sample space is non-uniform.
For large value of independent trials n: When number of independent trials n -->infinity, binomial distribution function tends to be the normal distribution function.
Stirling's approximation: click