1. You have bowl of 20 marbles that are identical except for color, 10 are red and 10 are green. Which of the following are Bernoulli trials ?
(a) You blindly pick 10 marbles out in succession, determine for each pick whether the marble is red, and then replace the marble in the bowl after each pick.
(b) The same experiment as in (a) except you do not replace the marble after each pick.
2. AT&T claims that when customers call directory assistance for telephone numbers, the right number is given 90% of the time. Assume a 90% rate of correct responses and assume that we want to find the probability that among 5 requests, 3 of the responses are correct.
(a) Does this procedure result in a Binomial distribution ?
(b) If this procedure does result in a binomial distribution, identify the values of n, x, p and q.
3. A bowl contains 20 marbles that are identical except for color : 12 are red and 8 are green. You blindly pick 6 marbles from the bowl, returning each marble after its color has been observed. How many ways can you pick 4 red marbles in 6 trials ?
4. Use the Binomial probability formula to find the probability of getting exactly three correct responses among five different requests. Assume that in general it is correct 90% of the time. That is, find P(3) given that n=5, x=3, p=0.9 and q=0.1.
5. Suppose an experiment consist of three repetitions of a Bernoulli trial with the probability of success equal to one quarter and the probability of failure equal to three quarter. What is the probability that exactly one success is obtained in the three trials?
6. Each sample of air has a 10% chance of containing a particular rare molecule. Assume that the samples are independent with regard to the presence of the rare molecule. Find the probability that in the next 18 samples
(a) exactly two contain the rare molecule.
(b) at least four samples contain the rare molecule.
7. Suppose an experiment consist of three repetitions of a Bernoulli trial with the probability of success equal to one quarter and the probability of failure equal to three quarter. Solve the problem by using Binomial formula, then find the probability that
(a) exactly one success
(b) more than two successes
(c) at most two successes
8. Three fair coins are spun. Find the probability there is exactly one head ?
9. For a biased coin, the probability of landing heads up is one quarter. The coin is tossed three times. List the ways in which there could be one head (H) and two tails (T). Find the probability that there will be exactly one head.
10. Suppose that X is a random variable, with X~B(10, 0.3). Find the probability that X = 2.
11. There are 1000 students in a college and it is known that 200 of them study Mathematics. 10 students are picked at random. Use a Binomial distribution to find the approximate probability that the sample of 10 contains 4 Mathematics students.
12. Find the mean, variance and standard deviation of the Binomial probability distribution for the number of heads in three flips of a coin.
13. Given Binomial random variable X~B(20, 0.06), find P(X ≥ 4).
14. Random variable X is Binomial distributed with n=50 and p=0.08. By using Binomial probability table, find
(a) P(X ≥ 8)
(b) P(X < 4)
(c) P(3 ≤ X ≤ 9)
(d) P(8 < X ≤ 12)
(e) P(X = 7)
15. Let p = 0.25 and n = 6. Find the probability that
(a) exactly three are successes
(b) at most two are successes
(c) at least three are successes
(d) one to three is successes
16. The probability one games will end with success in 10 times is 0.3. Find the mean, variance and standard deviation for Binomial distribution which involved 1000 games ?
17. A man plants ten saplings. For each sapling the probability that it will grow into a mature tree is 0.25 and each sapling is independent of the others. Find the mean, variance and standard deviation of the number of mature trees he obtains.
18. A government inspector responsible for checking cars for defective tire finds a defective type in one out of every five cars examined. In a normal working week 80 cars are examined on average.
(a) What is the average number of cars per week found to have a defective tire ?
(b) What is the standard deviation of the number of cars per week with x defective tire ?
19. During a laboratory experiment the average number of radioactive particles passing through a counter in one millisecond is four. What is the probability that 6 particles enter the counter in a given millisecond ?
20. The mean number of bacteria per cm3 of a liquid is known to be three. Assuming that the number of bacteria follows a Poisson distribution, find the probability that, there will be
(a) no bacteria in 1 cm3 of liquid
(b) four bacteria in 1 cm3 of liquid
(c) less than three bacteria in 2 cm3 of liquid
(d) in half ml of liquid there will be more than two bacteria
21. Ten is the average number of oil tankers arriving each day at a certain port city. The facilities at the port can handle at most 15 tankers per day. What is the probability that on a given day tankers have to be turned away ?
22. It is known that two percent of the books bound at a certain bindery have defective bindings. Use the Poisson approximation to the Binomial distribution to find the probability that five of 400 books bound by this bindery will have defective bindings.
23. For the case of the thin copper wire, suppose that the number of flaws follows a Poisson distribution with a mean of 2.3 flaws per millimeter. Determine the probability of
(a) exactly two flaws in 1 millimeter of wire
(b) ten flaws in five millimeters of wire
(c) at least one flaws in 2 millimeters of wire
24. The number of daisies on a square meter of lawn has a Poisson distribution with mean five. Find the probability there are four daisies in a square meter.
25. Contamination is a problem in the manufacture of optical storage disks. The number of particles of contamination that occur on an optical disk has a Poisson distribution and the average number of particles per centimeter squared of media surface is 0.1. The area of a disk under study is 100 squared centimeters. Find the probability that 12 particles occur in the area of a disk under study.
26. The number of taxis that arrive at a hotel in each period of five minutes is modeled by a Poisson distribution with mean three. Find the probability that at least two taxis will arrive in a given five minute period.
27. The number of misspellings in a page of a newspaper has a Poisson distribution with mean 2.7. Find the probability that in a page
(a) There are three misspellings.
(b) There are fewer than two misspellings.
28. It is thought that the number of flaws on a meter of cloth produced by a machine follows a Poisson distribution. The proportion of flaws free meter is 0.3. Find the mean number of flaws.
29. Records show that the probability is 0.00006 that a car tire will go flat while being driven through a certain tunnel. Use the Poisson approximation to the Binomial distribution to find the probability that at least two of 10,000 cars passing through that tunnel will get flat tires.
30. A typist is copying a list of 1000 numbers. For each number, the probability that she will type it wrongly is 0.002. Use the Poisson approximation to the Binomial to find the probability that there will be fewer than two mistakes in the whole list.
31. A chain is made out of 1000 links, each of which has probability 0.0006 of breaking under tension. Find the probability that at least one link will break under tension.
32. On average, 2% of the items produced on a production line are defective. The items are packed in boxes of 100.Use the Poisson approximation to the Binomial to calculate the probability that a box contains
(a) no defective items
(b) one defective item
(c) at most two defective items
33. Random variable, X is Binomially distributed with n=50 and p=0.05. Evaluate P(X ≥ 2) by
(a) using Binomial distribution
(b) using Poisson approximation
34. Find the probability by using standard Normal table
(a) P(Z ≥ 0.5) (e) P( | Z | > 1)
(b) P(Z ≥ 1.0) (f) P(-1.96 ≤ Z ≤ 0)
(c) P(Z ≤ -1.86) (g) P(-3 ≤ Z ≤ -2)
(d) P(0 ≤ Z ≤ 2.2)
35. If given X~ N(1,4), find P(-3 ≤ X ≤ 3).
36. Find the area under the Normal distribution curve
(a) between Z = 0 and Z = 2.34.
(b) between Z = 0 and Z = 1.8.
(c) between Z = 0 and Z = -1.75.
(d) to the right of Z = 1.11.
(e) to the left of Z = -1.93.
(f) between Z = 2.00 and Z = 2.47.
(g) between Z = -2.48 and Z = -0.83.
(h) between Z = 1.68 and Z = -1.37.
(i) to the left of Z = 1.99.
(j) to the right of Z = -1.16.
(k) to the right of Z = 2.43 and to the left of Z = -3.01.
37. Find the probability for each
(a) P( 0 < Z < 2.32 )
(b) P( Z < 1.65 )
(c) P( Z > 1.91 )
38. In a large class, suppose your instructor tells you that you need to obtain a grade in the
top 10% of your class to get an A on a particular exam. From past experience she is able
to estimate that the mean and standard deviation on this exam will be 72 and 13,
respectively. What will be the minimum grade needed to obtain an A ? (Assume that the
grades will be approximately Normally distributed)
39. The incomes of junior executives in a large corporation are normally distributed with a
standard deviation of RM1200. A cutback is pending, at which time those who earn less
than RM28000 will be discharged. If such a cut represents 10% of the junior executives,
what is the current mean salary of the group of junior executives ?
40. A multiple choice test contains 100 questions, each of which has five possible answers.
Find the probability that a candidate who answers at random gets more than 24 correct.
41. An unnoticed mechanical failure has caused one third of a machine shop's production of
5000 rifle firing pins to be defective. What is the probability that an inspector will find no
more than three defective firing pins in a random sample of 25 ?