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.