Definition:
The Binomial Distribution is one of the discrete probability distribution.
It is used when there are exactly two mutually exclusive outcomes of a trial.
These outcomes are appropriately labeled Success and Failure. The Binomial Distribution is used to obtain
the probability of observing r successes in n trials, with the probability of success on a single trial denoted by p.
Formula:
P(X = r) = nCr p r (1-p)n-r
where,
n = Number of events.
r = Number of successful events.
p = Probability of success on a single trial.
nCr = ( n! / (n-r)! ) / r!
1-p = Probability of failure.
Example: Toss a coin for 12 times. What is the probability of getting exactly 7 heads.
Step 1: Here,
Number of trials n = 12
Number of success r = 7 (since we define getting a head as success)
Probability of success on any single trial p = 0.5
Step 2: To Calculate nCr formula is used.
nCr = ( n! / (n-r)! ) / r!
= ( 12! / (12-7)! ) / 7!
= ( 12! / 5! ) / 7!
= ( 479001600 / 120 ) / 5040
= ( 3991680 / 5040 )
= 792
Step 3: Find pr.
pr = 0.57
= 0.0078125
Step 4: To Find (1-p)n-r Calculate 1-p and n-r.
1-p = 1-0.5 = 0.5
n-r = 12-7 = 5
Step 5: Find (1-p)n-r.
= 0.55 = 0.03125
Step 6: Solve P(X = r) = nCr p r (1-p)n-r
= 792 × 0.0078125 × 0.03125
= 0.193359375
The probability of getting exactly 7 heads is 0.19
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