ENGI 3423 Probability and Statistics

Faculty of Engineering and Applied Science
2010 Fall

Problem Set 3

[Probability and counting techniques, Bayes’ Theorem]

To be completed by Monday 04 October, 2010.
Compare your solution with the files listed on the solution page.

  1. Evaluate the following manually (show your working):

    1. 10C3
    2. 10P3
    3. 11C2
    4. 200C198

  1. A truck is carrying six steel pipes and three copper pipes.
    A random sample of four pipes is taken [without replacement] from the truck.   Find:

    1. the number of ways there can be exactly two copper pipes in the sample.
    2. the number of ways there can be at least two copper pipes in the sample.

    Show your working.

  1. Just as the binomial coefficient

    nCr = n! / (r! (n-r)!)
    counts the number of distinct partitions of n items into two piles, one of r items and the other of the remaining (n – r) items, so the multinomial coefficient
    (n | r1 r2 ... rk) = n! / (r1! r2! ... rk!)
    counts the number of distinct partitions of n items into k piles, one of r1 items, another of r2 items, another of r3 items and so on, (where, in this case, r1 + r2 + r3 + ... + rk = n).
    [The binomial coefficient is then a special case of the multinomial coefficient, with k = 2, r1= r and r2= n – r.]

    1. Use the appropriate multinomial coefficient to find the number of distinct ways in which a manager can divide a work crew of twelve employees into groups of three, four and five people.

    2. Find the number of distinct ways of rearranging the letters in the word "STATISTICS" among themselves.

  1. A bag contains four tiles, each marked with a letter, one each of A B C D .

    1. If the four letters A B C D are drawn at random without replacement, what is the probability that the letter B will be drawn immediately after the letter A ?

    2. If the four letters A B C D are drawn at random with replacement, what is the probability that the letter B will be drawn immediately after the letter A ?

  1. One urn contains five red balls and five green balls.   A second urn contains eight red balls and one green ball.   A ball is chosen randomly from the first urn and placed in the second urn.   Then a ball is chosen randomly from the second urn and placed in the first urn.

    1. What is the probability that a red ball is selected from the first urn and a red ball is selected from the second urn?

    2. What is the [unconditional] probability that a red ball is selected from the second urn?

    3. At the conclusion of the selection process, what is the probability that the numbers of red and green balls in each urn are identical to the initial numbers?

    Now consider the generalization to the case where there are n (> 1) balls initially in each urn, with a (0 < a < n) red balls in urn 1 and b (0 < b < n) red balls in urn 2 initially.

    1. Show that the [unconditional] probability that a red ball is selected from the second urn during this process is, in general, different from the probability of drawing a red ball directly from the second urn [when the first urn is left alone].

    2. Find a necessary and sufficient condition on the values of a and b for the two unconditional probabilities in part (d) above to be equal.

  1. A factory has the following information about the quality control process on its production line:

    If an item is defective, then there is a 99.9% chance that the quality control process will reject it [which, with the benefit of hindsight, is the correct decision].
    If an item is good, then there is a 2% chance that the quality control process will reject it [a “false positive”, which, with the benefit of hindsight, is not the correct decision].
    It is known that 0.1% of all items on the production line are defective.

    Given that the quality control process has just rejected an item, find the probability that that decision is correct [i.e. that the item really is defective].

  1. A factory manager is trying to decide whether to invest in a new quality control process on a production line.

    If the manager decides against the investment, then a decision to accept or reject a product will be based only on her prior assessment of the probability that the product is defective, together with the consequences of each possible outcome:
            reject defective product (true positive),
            reject good product (false positive),
            accept defective product (false negative),
            accept good product (true negative).

    If the manager decides to invest in the new quality control process, then a decision to accept or reject a product will be based on the verdict of the quality control process, together with the consequences of each possible outcome and the cost of the investment.

    Draw a decision tree diagram that illustrates this situation.

    [Recall that decision nodes should be represented by squares and chance nodes by circles.   Be careful to place the nodes in the correct order.   We will develop this example further in problem set 4.]

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Created 2003 06 10 and most recently modified 2010 07 30 by Dr. G.H. George