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%

Binomial Distribution

To understand binomial distributions and binomial probability, it helps to understand binomial experiments and some associated notation; so we cover those topics first.

Binomial Experiment
A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties:

The experiment consists of n repeated trials.
Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
The probability of success, denoted by P, is the same on every trial.
The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.
Consider the following statistical experiment. You flip a coin 2 times and count the number of times the coin lands on heads. This is a binomial experiment because:

The experiment consists of repeated trials. We flip a coin 2 times.
Each trial can result in just two possible outcomes - heads or tails.
The probability of success is constant - 0.5 on every trial.
The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials.
Notation
The following notation is helpful, when we talk about binomial probability.

x: The number of successes that result from the binomial experiment.
n: The number of trials in the binomial experiment.
P: The probability of success on an individual trial.
Q: The probability of failure on an individual trial. (This is equal to 1 - P.)
b(x; n, P): Binomial probability - the probability that an n-trial binomial experiment results in exactly x successes, when the probability of success on an individual trial is P.
nCr: The number of combinations of n things, taken r at a time.
Binomial Distribution
A binomial random variable is the number of successes x in n repeated trials of a binomial experiment. The probability distribution of a binomial random variable is called a binomial distribution (also known as a Bernoulli distribution).

Suppose we flip a coin two times and count the number of heads (successes). The binomial random variable is the number of heads, which can take on values of 0, 1, or 2. The binomial distribution is presented below.

Number of heads Probability
0 0.25
1 0.50
2 0.25
The binomial distribution has the following properties:

The mean of the distribution (μx) is equal to n * P .
The variance (σ2x) is n * P * ( 1 - P ).
The standard deviation (σx) is sqrt[ n * P * ( 1 - P ) ].
Binomial Probability
The binomial probability refers to the probability that a binomial experiment results in exactly x successes. For example, in the above table, we see that the binomial probability of getting exactly one head in two coin flips is 0.50.

Given x, n, and P, we can compute the binomial probability based on the following formula:

Binomial Formula. Suppose a binomial experiment consists of n trials and results in x successes. If the probability of success on an individual trial is P, then the binomial probability is:
b(x; n, P) = nCx * Px * (1 - P)n - x


Example 1

Suppose a die is tossed 5 times. What is the probability of getting exactly 2 fours?

Solution: This is a binomial experiment in which the number of trials is equal to 5, the number of successes is equal to 2, and the probability of success on a single trial is 1/6 or about 0.167. Therefore, the binomial probability is:

b(2; 5, 0.167) = 5C2 * (0.167)2 * (0.833)3
b(2; 5, 0.167) = 0.161

Example 1

What is the probability of obtaining 45 or fewer heads in 100 tosses of a coin?

Solution: To solve this problem, we compute 46 individual probabilities, using the binomial formula. The sum of all these probabilities is the answer we seek. Thus,

b(x < 45; 100, 0.5) = b(x = 0; 100, 0.5) + b(x = 1; 100, 0.5) + . . . + b(x = 45; 100, 0.5)
b(x < 45; 100, 0.5) = 0.184


Example 2

The probability that a student is accepted to a prestigeous college is 0.3. If 5 students from the same school apply, what is the probability that at most 2 are accepted?

Solution: To solve this problem, we compute 3 individual probabilities, using the binomial formula. The sum of all these probabilities is the answer we seek. Thus,

b(x < 2; 5, 0.3) = b(x = 0; 5, 0.3) + b(x = 1; 5, 0.3) + b(x = 2; 5, 0.3)
b(x < 2; 5, 0.3) = 0.1681 + 0.3601 + 0.3087
b(x < 2; 5, 0.3) = 0.8369


Example 3

What is the probability that the world series will last 4 games? 5 games? 6 games? 7 games? Assume that the teams are evenly matched.

Solution: This is a very tricky application of the binomial distribution. If you can follow the logic of this solution, you have a good understanding of the material covered in the tutorial, to this point.

In the world series, there are two baseball teams. The series ends when the winning team wins 4 games. Therefore, we define a success as a win by the team that ultimately becomes the world series champion.

For the purpose of this analysis, we assume that the teams are evenly matched. Therefore, the probability that a particular team wins a particular game is 0.5.

Let's look first at the simplest case. What is the probability that the series lasts only 4 games. This can occur if one team wins the first 4 games. The probability of the National League team winning 4 games in a row is:

b(4; 4, 0.5) = 4C4 * (0.5)4 * (0.5)0 = 0.0625

Similarly, when we compute the probability of the American League team winning 4 games in a row, we find that it is also 0.0625. Therefore, probability that the series ends in four games would be 0.0625 + 0.0625 = 0.125; since the series would end if either the American or National League team won 4 games in a row.

Now let's tackle the question of finding probability that the world series ends in 5 games. The trick in finding this solution is to recognize that the series can only end in 5 games, if one team has won 3 out of the first 4 games. So let's first find the probability that the American League team wins exactly 3 of the first 4 games.

b(3; 4, 0.5) = 4C3 * (0.5)3 * (0.5)1 = 0.25

Okay, here comes some more tricky stuff, so listen up. Given that the American League team has won 3 of the first 4 games, the American League team has a 50/50 chance of winning the fifth game to end the series. Therefore, the probability of the American League team winning the series in 5 games is 0.25 * 0.50 = 0.125. Since the National League team could also win the series in 5 games, the probability that the series ends in 5 games would be 0.125 + 0.125 = 0.25.

The rest of the problem would be solved in the same way. You should find that the probability of the series ending in 6 games is 0.3125; and the probability of the series ending in 7 games is also 0.3125.