Exercise 1.7 (Binomial distribution)

EXERCISE 1.7

1. If the chance that any one of the 10 telephone lines is busy at any instant is 0.2, what is the chance that 5 of the lines are busy? What is the probability that all the lines are busy? 

[Ans. 10C₅ (0.2)⁵ (0.8)⁵; (0.2)¹⁰]

2. 20% of the bolts produced by a machine are defective, determine the probability that out of 4 bolts chosen at random. (a) 1 (b) 0, (c) Atmost 2 bolts, will be defective. 

[Ans. (a) 0.4096; (b) 0.4096 ; (c) 0.9728]

3. If the probability that a man aged 60 will live to be 70 is 0.65, what is the probability that out of 10 men, now 60, atleast 7 will live to be 70 ?  [Ans. 0.509]

4. If on the average rain falls on 10 days in every 30 days, obtain the probability that (i) rain will fall on atleast 3 days of a given week (ii) first three days of a given week will be fine and the remaining 4 days wet. 

[Ans. (i) 0.4294; (ii) 0.0037]

5. Fit a binomial distribution for the following data:

[Ans. (0.432 + 0.568)⁵]

6. If the probability of hitting a target is 10% and 10 shots are fired independently, what is the probability that the target will be hit atleast once ? 

[Ans. 65 nearly]

7. Assume that half of the population is vegetarian so that the chance of an individual being a vegetarian is 1/2. Assuming that 100 investigators take samples of 10 individual each to see whether they are vegetarian, how many investigators would you expect to report that three people or less were vegetarians? 

[Ans. 17]

8. In a certain town, 20% samples of the population is literate and assume that 200 investigators take samples of ten individuals to see whether they are literate. How many investigators would you expect to report that 3 people or less are literates in the samples? 

[Ans. 176]

9. Five fair coins are flipped. If the outcomes are assumed independent, find the probability mass function of the number of heads obtained.

10. It is known that screws produced by a certain company will be defective. with probability 0.01 independently of each other. The company sells the screws in packages of 10 and offers a money-back guarantee that atmost 1 of the 10 screws is defective. What proportion of packages sold must the company replace? 

[Ans. 0.005]

11. Suppose that the random variable X is equal to the number of hits [eo obtained by a certain baseball player in his next 3 bats. If P(X = 1) = 0.3, P(X = 2) = 0.2, and P(X = 0) = 3 P(X = 3). Find E (X). 

[Ans. 1.075]