In: Statistics and Probability
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)
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 px qn-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