ENGI 3423 Probability and Statistics

Faculty of Engineering and Applied Science
2010 Fall

Problem Set 2

[Elementary probability and Venn diagrams]

To be completed by Monday 27 September, 2010.
Compare your solutions with those in the files listed on the solution page.

1. Draw a decision tree, involving choice and chance forks, for each of

   1. a personal decision (for example, renewing a mortgage - what term?).
   2. an engineering problem.

2. 1. An engineer states that the odds of a reinforced girder surviving an average load of 2 MN for 25 years is “7 to 1 on”.   What is the engineer’s probability for this event?

   2. The event E is defined to be
            “the score on a roll of a fair die is an odd prime number”;
            (note:   the smallest prime number is 2).
      Calculate the odds on E, expressed as a ratio reduced to its lowest terms.

3. Odds vs. Bookmaker’s Odds

   Five teams are competing in a design competition which exactly one of them will win.   An observer places a non-refundable deposit of $1 for the opportunity of receiving a reward for a correct prediction of which team will win.   The reward is based upon the odds quoted here:

   Team A :	5:3 on
   Team B :	13:7 against
   Team C :	4:1 against
   Team D :	19:1 against
   Team E :	39:1 against

   1. Find the probabilities associated with the odds quoted above.
   2. Show that the probabilities associated with these odds are not coherent.
   3. For each event, find the reward for a correct prediction (that that team will win).
   4. If 25 observers pay a deposit and predict that team A will win, 14 predict B, 8 predict C, 2 predict D and 1 predicts E, then find the bookmaker’s profit for each of the five outcomes.
   5. Now assume that the relative chances of victory for each team are reflected by the relative odds quoted above, (so that, for example, team C really is four times as likely to win as team D).   Rescale the five odds quoted above so that the bet becomes a fair bet.
   6. Under this fair bet, what now is the bookmaker’s profit for each of the five outcomes?
   7. Comment on the relationship between the distribution of profits and the distribution of the numbers of deposits.

4. For the sample space S which consists of the first ten natural numbers { 1, 2, ... , 10 }, events A, B, C are defined as
   

   A = { odd numbers in S }
   B = { perfect square numbers in S }
   C = { 3, 5 }

   [A “perfect square” is a whole number whose square root is also a whole number.]
   
   1. Represent the above sets in a Venn diagram.
   2. Express the set D = { odd perfect square numbers in S } as a set function of the above sets.
      How many elements are in this set?
   3. Simplify the expression   A ∩ C .
   4. Simplify the expression   A ∪ C .
   5. Simplify the expression   B ∩ C .
   6. What elements are in the set   ~A ∩ ~B ∩ ~ C ?

5. In the quality control of mass-produced ceramic tiles,
   the probability that the next tile checked by an inspector has an air bubble is .05000,
   the probability that it is below acceptable mass is .04000,
   the probability that it has a crack is .02000,
   the probability that it either has an air bubble or is below acceptable mass or both is .08800,
   the probability that it is either below acceptable mass or cracked or both is .05880,
   the probability that it either is cracked or has an air bubble or both is .06800 and
   the probability that it has at least one of these imperfections is .10484.

   1. What is the probability that the next tile checked by the inspector has none of these imperfections?
   2. What is the probability that the next tile checked by the inspector has an air bubble as its only imperfection?
   3. What is the probability that the next tile checked by the inspector has an air bubble and is cracked?
   4. What is the probability that the next tile checked by the inspector has all of these imperfections?
   5. Are these three types of imperfection mutually exclusive?
   6. Are these three types of imperfection [stochastically] independent?

6. A leading engineering company presently has bids out on three projects.
   Let   A = { project 1 is awarded to the company },
           B = { project 2 is awarded to the company }
   and   C = { project 3 is awarded to the company }
   and suppose that
   P[A] = .86,     P[B] = .79,     P[C] = .58,
   P[A ∧ B] = .70,     P[B ∧ C] = .49,     P[C ∧ A] = .50
   and P[A ∧ B ∧ C] = .45.

   1. Draw a Venn diagram that represents this situation.
   2. Find the probability that the company wins exactly one of the three bids.

   Express in words each of the following events and compute the probability of each event:

   3.    ~A ∧ ~B
   4.    A ∨ B ∨ C
   5.    ~A ∧ ~B ∧ ~C
   6.    A ∧ B ∧ ~C
   7.    ~A ∧ ~B ∧ C

7. In a competition to award two identical contracts, five firms are tied with the lowest acceptable tender.   It is decided to award the contracts to exactly two of the five firms by a random process, so that each pair of firms has an equal chance of being chosen.

   The five firms have 4, 11, 12, 15 and 19 complete years’ experience respectively.
   [Note that two firms with 5 years 11 months’ experience would each have only five complete years and a total between them of only ten complete years, not eleven or twelve.   The quantity “years” in this question is therefore a discrete quantity, taking on non-negative integer values only.]
   Find the probability that the two chosen firms have a total of

   1. at least 25 complete years’ experience.
   2. more than 30 complete years’ experience.

8. [BONUS QUESTION]

   Three identical boxes each contain two marbles.   One box contains two red marbles, another box contains two blue marbles and the remaining box contains one of each.   All three boxes are closed and are labelled incorrectly.

   You are permitted to select any box and to withdraw one marble (without replacement) from that box.   You may repeat this process of selecting any box and withdrawing a marble without replacement as many times as necessary, until you are able to deduce the true contents of all three boxes.

   What is the least number of marbles that you must withdraw in order to be sure of identifying the contents of all three boxes correctly?

   Justify your answer with full working.

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