APPLICATION PROBLEMS ON BINOMIAL DISTRIBUTION

Problem 1 :

A coin is tossed 10 times. Assuming the coin to be unbiased, what is the probability of getting

i) 4 heads ?

ii) at least 4 heads ?

iii) at most 3 heads ?

Solution :

Let X be the random variable representing number of heads.

Number of trails (n) = 10

probability of getting head when a coin is tossed = 1/2

probability of not getting head when a coin is tossed = 1/2

p = 1/2 and q = 1/2

i)  

ii) at least 4 heads :

P(x  ≥ 4) = 1 - P(x < 4)

= 1 - [P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3)]

iii)  at most 3 heads :

Problem 2 :

If 15 dates are selected at random, what is the probability of getting two Sundays ?

Solution :

Number of days selecting (n) = 15

Probability of getting two Sundays (p) = 1/7

Probability of getting failure (q) = 6/7

Problem 3 :

The incidence of occupational disease in an industry is such that the workmen have a 10% chance of suffering from it. What is the probability that out of 5 workmen 3 or more will contract the disease ?

Solution :

Probability that the workmen suffers from the occupational disease p = 0.1 = 1/10

Probability does not suffer q = 0.9 = 9/10 

Number of workmen choosing n = 5

= P(x ≥ 3)

Problem 4 :

6 coins are tossed 512 times. Find the expected frequency of heads. Also compute the mean and standard deviation of the number number of heads.

Solution :

Number of coins tossed (n) = 6

Probability of getting heads = 1/2

not getting heads q = 1/2

Getting 0 heads :

Getting 1 head :

Getting 2 heads :

Getting 3 heads :

Getting 4 heads :

Getting 5 heads :

Getting 6 heads :

Expected Frequency :

Mean :

Mean = np

= 6 (1/2)

= 3

Standard deviation :

Variance = npq

Standard deviation = √npq

= √6x(1/2)x(1/2)

= √1.5

= 1.22