Question

How can you demonstrate that the probability of X successes in a binomial distribution is given by P(X=x)=nCx p^x q^(n-x)

Answer 1

Solution:

To calculate P(x) you need to know two things :

1. How many combinations of outcomes would provide x number of successes, nCx.
2. The probability of a success in any given trial (p)

Formula: nCx = n! / (n - x)! x!

For example:

4C2 = 4! / (2! 2!) = 24 / 4 = 6

Now we take a simple example to estimate the probability of total number of x successes P(x) of Binomial distribution for

n = 18, X = 12 and P = 0.36

Step 1. Number of trials (n) = 18
Number of success (x) = 12
Success probability for each trial p = 0.36
P(x) = nCx p^x q^n-x

Step 2.

18C12 =18!12!(18-12)!
=18!12! x 6!
18C12 = 18564

Step 3. To find value of q

q = 1 - p
q = 1 - 0.36
q = 0.64

Step 4. Put in formula:

P(x) = 18564 x 0.36¹² x 0.64⁶

Step 5. Solve expression

P(x) = 18564 x 0 x 0.0687
P(x) = 0.006

Hence 0.006 is the binomial probability of getting 12 successes from 18 total number of events with 0.36 success probability for each trial.

Done