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Binomial Distribution

  1.  If a random variable X takes two values 1 and 0 with probability p and q respectively and q = 1 - p then this is called Bernoulli distribution. Here p is called the probability of success and q is called the probability of failure.

For n trials, the probability of x successes (x≤ n) is given by Binomial distribution. The probability mass function is defined as follows :

n = No. of independent trials

x = No. of successes

p = Probability of success on any given trial

q= 1- p

Generally, it is denoted as B(n, p).

 

II. Properties

(i)                 Distribution function

(iii) First two moments about origin.

= n(n -1) p2. (q+p)n-2 + np

= n(n- 1) p2 + np.

Now,               µ1=µ1=np,which is the mean

µ2=µ2-(µ1)2

=n( n-1)p2 + np – n2p2

=np-np2

=np(1-p)which is the variance.

Similarly we obtain µ3 and µ4

(iv) Skewness:
( v) Kurtosis : 

(vi) Mode is the value of x for which P[X = n] is_ maximum.

When (n + 1) p is not an integer,

Mode = Integral part of (n + 1)p.

When (n + 1) p is an integer, we obtain two modes

Mode = (n + 1) p and (n + 1) p - 1. ·

 

Example 1. Determine the binomial distribution for which the mean is 8 and variance 4 and find its mode.

Solution. Given that np = 8 and npq = 4

On division, we get q = ½                              =>        p =  l – q  = 1/2

Also n = 8/p = 16

Thus the given binomial distribution is B(16,1/2).

Now, (n + 1) p = (16 + 1) ½=17/2=8+1/2

which implies that mode= 8 (integral part only).

 

Example 2. The mean and variance of a binomial distribution are 5 and 2 respectively. Find P[X -1].

Solution. Given that   np = 5,                        npq = 5/2

On division we get      q = 1/2,           p = 1/2

Also,                            n = 5/2 = 10

P[X - 1] = P[X = 0] + P[X = 1]

Example 3. In a shooting competition, the probability of a man hitting a target is 215. If he fires 5 times, what is the probability of hitting the target (i) at least_ twice (ii) at most twice.

Solution. Let p = hitting a target = 2/5. q = 1 - p =3/5. n = 5

(i)                 P[at least twice hitting] = 1 - [P(no hitting) + P(one hitting)]

=1 - 0.337 = 0.66

(i)                 P[ at most twice hitting)= P(no hitting) + P( one hitting) + P(two hitting)

=0.68.

 

Example 4. If 4 of 12 scooterists do not carry driving licence, what is the probability that a traffic inspector who randomly selects 4 scooterists, will catch

(i) I for not carrying driving licence.

(ii) at least 2 for not carrying driving licence.

 

Solution.         P= Probability that a scooterist does not carry driving licence

=4/12=1/3        

         q = 1 - p = 2/3, n = 4

(i)                 P (catching one scooterist having no driving licence)

(ii) P (catching at least two scooterists having no driving licence)

 

=1 - [P(all having licence) + P(l having no licence)]

Example 5. Fit a binomial distribution to the following distribution :

Solution.                     Mean=∑ x f / ∑ f= 35/50 , n=5

Therefore,        np =35/ 50  =>  p 35/250  =0.14   and   q=  0.86

The expected frequencies of the fitted binomial distribution can be calculated from

50 (0.86 + 0.14)5

 

PROBLEMS

  1. Five prizes are to be distributed among 20 students. Find the probability that a particular student will receive three prizes.
  2. Consider an intersection approach in which studies have shown 25% right turns and no left turns. Find the probability of one out of the next four vehicles turning right.
  3. The mean and variance of a binomial distribution are 6 and 2 respectively. Find P[X > 1],P[X = 2].
  4. In a binomial distribution consisting of 5 independent trials, probabilities of I and 2 successes are 0.4096 and 0.2048. Find the parameter p of the distribution.
  5. An experiment succeeds twice as often as it fails. What is the probability that in next five trials there will be (i) three successes, (ii) at least three successes ?
  6. A quality control engineer inspects a random sample of 3 calculators from each lot of 20 calculators.

8 .If such a lot contains 4 slight defective calculators. What are the probabilities that the inspector's sample will contain

(i) no slight defective calculators,

(ii) one slight defective calculators,

(iii) at least two slight defective calculators.

 

9. Fit a binomial distribution to the following distribution

9. Fit a binomial distribution to the following data.

 

10. How many tosses of a coin are needed so that the probability of getting at least one head is 87.5% ?

11. A machine produces an average of 20% defective bolts. A batch is accepted if a sample of 5 bolts taken from that batch contains no defective and rejected it the sample contains 3 or more defectives. In other cases, a second sample is taken. What is the probability that the second sample is required ?

12. If the probability of a defective bolt is 0.1, find (i) mean, (ii) variance, (iii) moment coefficient of skewness and, (iv) Kurtosis for the distribution of defective bolts in a total of 400.

13. If on an average I vessel in every 10 is wrecked, find the probability that out of 5 vessels expected to arrive, at least 4 will arrive safely.

ANSWERS

  1.  0.088                                      2          . 0.56                           3.         0.999, 0.007

4. p =1/5                                       5.         (i) 80/243                    (ii)192/243

6. (i)64/125            (ii) 48/125                   (iii)13/125

7.