Binomial Distribution

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 Successand 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) = _nC_r p ^r (1-p)^n-r
where,
             n = Number of events
             r = Number of successful events.
             p = Probability of success on a single trial.
             _nC_r = ( 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 _nC_r formula is used. 
           _nC_r = ( n! / (n-r)! ) / r!
                = ( 12! / (12-7)! ) / 7!
                = ( 12! / 5! ) / 7!
                = ( 479001600 / 120 ) / 5040
                = ( 3991680 / 5040 )
                = 792

  Step 3: Find p^r.
            p^r = 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) = _nC_r p ^r (1-p)^n-r
            = 792 * 0.0078125 * 0.03125
            = 0.193359375


The probability of getting exactly 7 heads is 0.19

Binomial Distribution Problems

(1)  A company owns 400 laptops.  Each laptop has an 8% probability of not working.  You randomly select 20 laptops for your salespeople.

(a) What is the likelihood that 5 will be broken?         (b) What is the likelihood that they will all work?

(c) What is the likelihood that they will all be broken?

(2) A study indicates that 4% of American teenagers have tattoos.   You randomly sample 30 teenagers.   What is the likelihood that exactly 3 will have a tattoo?

(3)  An XYZ cell phone is made from 55 components.  Each component has a .002 probability of being defective. What is the probability that an XYZ  cell phone will not work perfectly?

(4)  The ABC Company manufactures toy robots.    About 1 toy robot per 100 does not work.  You purchase 35 ABC toy robots. What is the probability that exactly 4 do not work?

(5)  The LMB Company manufactures tires.  They claim that only .007 of LMB tires are defective.  What is the probability of finding 2 defective tires in a random sample of 50 LMB tires?

(6)  An HDTV is made from 100 components.  Each component has a .005 probability of being defective. What is the probability that an HDTV will not work perfectly?

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(1) (a)  20C5 (.08)⁵ (.92)¹⁵ = .0145   (b) 20C0 (.08)⁰(.92)²⁰ = .1887 
(c) 20C20 (.08)²⁰(.92)⁰ = .0000000000000000000001 (note ^-22 means move the decimal 22 places to the left)

(2) 30C3 (.04)³ (.96)²⁷ = .0863

(3) Probability that it will work (0 defective components) 55C0 (.002)⁰ (.998)⁵⁵ = .896

Probability that it will not work perfectly is 1 - .896 = .104   or 10.4%

(4) 35C4 (.01)⁴ (.99)³¹ = .00038

(5) 50C2 (.007)² (.993)⁴⁸ = .0428 
(6) Probability that it will work (0 defective components) 100C0 (.005)⁰ (.995)¹⁰⁰ = .606

Probability that it will not work perfectly is 1 - .606 = .394   or 39.40%