Miscellaneous Problems

Miscellaneous Problems

1. A manufacturer of metal pistons finds that on the average, 12% of his pistons are rejected because they are either oversize or undersize. What is the probability that a batch of 10 pistons will contain

(a) no more than 2 rejects? (b) at least 2 rejects?

2. Hospital records show that of patients suffering from a certain disease 75% die of it. What is the probability that of 6 randomly selected patients, 4 will recover?

3. If electricity power failures occur according to a Poisson distribution with an average of 3 failures every twenty weeks, calculate the probability that there will not be more than one failure during a particular week.

4. Vehicles pass through a junction on a busy road at an average rate of 300 per hour.

i. Find the probability that none passes in a given minute.

ii. What is the expected number passing in two minutes?

5. Entry to a certain University is determined by a national test. The scores on this test are normally distributed with a mean of 500 and a standard deviation of 100. Raghul wants to be admitted to this university and he knows that he must score better than at least 70% of the students who took the test. Raghul takes the test and scores 585. Will he be admitted to this university?

6. The time taken to assemble a car in a certain plant is a random variable having a normal distribution of 20 hours and a standard deviation of 2 hours. What is the probability that a car can be assembled at this plant in a period of time .

a) less than 19.5 hours?

b) between 20 and 22 hours?

7. The annual salaries of employees in a large company are approximately normally distributed with a mean of $50,000 and a standard deviation of $20,000.

(a) What percent of people earn less than $40,000?

(b) What percent of people earn between $45,000 and $65,000?

(c) What percent of people earn more than $70,00

8. X is a normally normally distributed variable with mean μ = 30 and standard deviation σ = 4. Find

(a) P(x < 40)

(b) P(x > 21)

(c) P(30 < x < 35)

9. The birth weight of babies is Normally distributed with mean 3,500g and standard deviation 500g. What is the probability that a baby is born that weighs less than 3,100g?

People’s monthly electric bills in chennai are normally distributed with a mean of ₹ 225 and a standard deviation of ₹ 55. Those people spend a lot of time online. In a group of 500 customers, how many would we expect to have a bill that is ₹ 100 or less?

Answers:

1. (i) 0.89131 (ii) 0.34173

2. 0.03295

3. 0.98981

4. 0.0067379 or 6.7379 ×10^-3

5. 80.33%

6. a) 0.4013 (b) 0.3413

7. a) 30.85% b) 37.20% c) 10.56%

8. a) 0.9938 (b) 0.9878 (c) 0.3944

9. 0.2119

10. 7