# Negative Binomial Summary for n = 15, r = 3 & P = 0.5 Negative binomial probability function solved example work with steps & calculation summary to estimate probability of total number of failures P(r) for number of failures X = 3, success probability for each trial P = 0.5 from the total number of events n = 15.

Calculation Summary
Number of Possibility (n)15
Number of Failures (r)3
Trial Probability (P)0.5
Negative Binomial distribution0.0026

## Work with Steps for n = 15, r = 3 & P = 0.5

The below is the complete work with step by step calculation for n = 15 , r = 3, p = 0.5 to estimate the probability of total number of failures in statistical experiments.
Workout :
step 1 Address the formula input parameters and values
Number of Possibility (n) = 15
Number of Failures (r) = 3
Probability (P) = 0.5
P(x)= r+n-1Cr Pn qr

step 2 Find value for 17 C 3 or 17 CHOOSE 3
17C3 = 17!/3!(17-3)!
= 17!/3! x 14!
= 355687428096000/6 x 87178291200
17C3 = 680

step 3 Find q value from P
q = 1 - P
q = 1 - 0.5
q = 0.5

step 4 Apply 17C3, P & q values in Negative binomial distribution formula
P(x) = 680 x 0.515 x 0.53

step 5 solve the expression
= 680 x 0.515 x 0.53
= 680 x 0 x 0.125
P(x) = 0.0026

Thus 0.0026 is the probability of total number of 3 failures from 15 events with 0.5 failure probability for each trial 