Definition:
The probability distribution of a Negative Binomial random variable is called a Negative Binomial Distribution. It is also known as the Pascal distribution or Polya distribution.
Suppose we flip a coin repeatedly and count the number of heads (successes). If we continue flipping the coin until it has landed 2 times on heads, we are conducting a
Negative Binomial Experiment.
Formula:
P(X = r) = n-1Cr-1 p r (1-p)n-r
where,
n = Number of events.
r = Number of successful events.
p = Probability of success on a single trial.
n-1Cr-1 = ( (n-1)! / ((n-1)-(r-1))! ) / (r-1)!
1-p = Probability of failure.
Example: Find the probability that a man flipping a coin gets the fourth head on the ninth flip.
Step 1: Here,
Number of trials n = 9 (because we flip the coin nine times).
Number of successes r = 4 (since we define Heads as a success).
Probability of success for any coin flip p = 0.5
Step 2: Find n-1 and r-1.
n-1 = 9-1 = 8
r-1 = 4-1 = 3
Step 3: To find n-1Cr-1 Calculate ((n-1)-(r-1))!
(n-1)-(r-1) = 8-3 = 5
((n-1)-(r-1))! = 5! = 120
Step 4: Find (n-1)!
= 8! = 40320
Step 5: Find (r-1)!
= 3! = 6
Step 6: Find (n-1)! / ((n-1)-(r-1))!
= 40320/120 = 336
Step 7: To Solve n-1Cr-1 formula is used.
= 336/6 = 56
Step 8: Find pr.
= 0.54 = 0.0625
Step 9: To Find (1-p)n-r Calculate 1-p and n-r.
1-p = 1-0.5 = 0.5
n-r = 9-4 = 5
Step 10: Calculate (1-p)n-r.
= 0.55 = 0.03125
Step 11: Calculate Negative Binomial Distribution.
= 56×0.0625×0.03125 = 0.109375
The probability that the coin will land on heads for the fourth time on the ninth coin flip is 0.1094.
|
| 
