ENGI 3423 Probability and Statistics

Faculty of Engineering and Applied Science
2007 Fall

Term Test 1

2007 October 03
[Descriptive Statistics, Elementary Probability, Bayes’ Theorem]

1. Fifty (50) observations of the pressure (in tens of kiloPascals) in prototype gas pipes are summarized in this relative frequency histogram:

   

   [Note:   no working is required in parts (a) and (b) of this question.]

   1. Identify the modal class for these data.

      [2]

   2. Does the histogram provide evidence for positive skew, negative skew or no skew?

      [2]

   3. Given the summary statistics , calculate the sample standard deviation s, correct to three significant figures.

      [4]

   4. Use the histogram to find the number of observations in the interval [4.0, 6.0).

      [4]

2. A broker quotes odds for a pair of complementary events A and B as
                 r_A = 4:1 on   and   r_B = 3:2 against.

   1. Calculate the corresponding probabilities   p_A   and   p_B.

      [4]

   2. Show that these probabilities are not coherent.

      [2]

   BONUS QUESTION:
   3. If a deposit of $100 is placed with the broker on each of events A and B, what is the broker’s profit (or loss) if event B occurs?

      [+4]

3. Events A, B, C are such that
          

   1. Use a Venn diagram to find the probability that all three events occur.

      [7]

   2. Are the three events mutually independent?   Show your working.

      [5]

4. Two otherwise identical unlabelled boxes contain different numbers of resistors and capacitors.   One box contains eight resistors and two capacitors.   The other box contains five resistors and five capacitors.   Upon selecting a box at random and withdrawing one item from that box at random, you find that the item is a resistor.   Express, as a fraction in its lowest terms, the probability that the box that you selected is the one that contained eight resistors, given that the withdrawn item is a resistor.

   [10]