Binomial Distribution

Binomial Distribution is the probability distribution of the number of successes in a sequence of n independent yes, no experiments each of which has a success with probability p. This is also known as Bernoulli Trial or experiment. The equation for this is nCr x p(s)^r x p(f)^n-r.

n=sample size

r=no. of successes

p(s)=probability of one success

p(f)=probability of one failure.

 A worked example of this is: in a large town one person in five is left-handed. Find the probability that in a class of thirty children exactly five are left-handed.

The answer is: 30C3 x 0.2³ x 0.8²⁷=0.0875.

30=sample size

3=number of successes (lefties)

0.2=probability of one success (as 1 in 5 are left-handed)

0.8=probability of one failure (as 4 in 5 are right-handed).

Sometimes you will be asked to find sample size, a worked example of this is:

 How large must a sample be if the probability of at least one left-handed person is to be greater than 90%.
p(s ≥ 1) ≥ 0.9
is the same as 1-p(s=0) ≥ 0.9
1-(nC0 x 0.8ⁿ x 0.2⁰ ≥ 0.9 = 1-0.8ⁿ ≥ 0.9
0.1 ≥ 0.8ⁿ
Trial and Error:

 0.8¹⁰=0.107 =failure
0.8¹¹=0.086 =success
n must be  ≥ 11

Notice that instead of writing out f=1 throught to f=30 we just did 1-f=0 which is much quicker.

Bionomial Expansion and Pascal's Triangle

Bionomial Expansion describes the algebraic expansion of powers of a binomial. A binomial is the sum of two monomials; algebraic expression consisting of a single term such as x.

The most basic formula of binomial expansion is (x+y)². This can be found using Pascal's triangle which gives the coefficient of the x and y values.

(x+y)²=x²+2xy+y².

Notice that in the first term there is no y value i.e y⁰.

From that point which each term the power of x decreases by 1 and the power of y increases by 1. Although Pascal’s triangle is attributed to Blaise Pascal who described them in the 17th century this special case of binomial expansion was first spotted by the 4th century BC Greek mathematician Euclid.

For a binomial involving subtraction Pascal's triangle can still be applied as long as the opposite of the second term is used. This makes every other term negative;

Pascal's Triangle