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Exam 2 Review

Directions. The following are review problems for the section. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. 
  1. An experiment consists of rolling a fair, four-sided die and observing the number that lands uppermost. If the result is an even number, we draw a card from a standard well-shuffled 52 card deck and record the color. If the result is odd, we flip a fair coin and record whether the side that lands uppermost is heads or tails. 
    Let \(H = \) heads, \(T= \)tails, \(R= \) Red, and \(B= \) Black
    1. Find an appropriate sample space for this experiment.
    2. Write (or list) the  events:
      1. ​\(E\) is the event a red card is drawn.
      2.  \(F\) is the event a tails occurs.​​
      3. \(G\) is the event an even number is rolled.
    3. ​List and describe the event \(E \cup F^C \cap G\).​

      1. \(S=\{(1,H), (1, T), (2, R), (2, B), (3, H), (3, T), (4, R), (4,B)\}\)
      2.  
        1. \(E=\{(2, R), (4, R)\}\)
        2. \(F=\{(1, T), (3, T)\}\)
        3. \(G=\{(2, R), (2, B), (4, R), (4, B)\}\)
      3. ​\(E \cup F^C \cap G= G\)

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      MATH 140 WIR6 #1


  2. Before Thanksgiving a group of Aggies were surveyed and asked to choose their favorite side
    dish at Thanksgiving dinner. The survey results are displayed in the table below.
      Candied
    Yams
    Corn
    Bread
    Dressing
    Roasted
    Brussels
    Sprouts
    Green
    Beans
    Other Total
    Freshman 5 20 30 10 20 85
    Sophomore 7 14 17 11 6 55
    Junior 2 32 25 20 10 89
    Senior 7 19 30 7 8 71
    Total 21 85 102 48 44 300
    If one Aggie is randomly selected from this group, what is the probability that the Aggie:
    1. likes roasted Brussels sprouts?
    2. is a freshman?
    3. is a sophomore whose favorite side dish was green beans?
    4. is a senior or junior whose favorite side dish was candied yams?​​

      1. \(\dfrac{102}{300}\)
      2. \(\dfrac{85}{300}\)
      3. \(\dfrac{11}{300}\)
      4. \(\dfrac{89+71+5+7}{300} = \dfrac{172}{300}\)

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      MATH 140 WIR6 #2


  3. Let \(E\) and \(F\) be two events of an experiment with sample space \(S\). Suppose that \(P(E) = 0.4, P(F) = 0.3,\) and \(P(E \cup F) = 0.6\). Find \(P(E^C \cap F)\). ​​

    \(P(E^C \cap F) = 0.2\)

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    MATH 140 WIR6 #3


  4. An experiment has a sample space of \(S = \{i,\  l,\  o,\  v,\  e,\  m,\  a,\  t,\  h\}\), with events \(E = \{t, a, i, l\} \), \(F = \{l,\  o,\  v,\  e\}\), and \(G=\{m,\ a,\ t,\ h\}\).  Find each of the following.
    1. \(E \cup F^C\)
    2. \(E \cap F^C\)
    3. \((F \cap G)^C\)​

      1. \(E \cup F^C = \{t, a, i, l, m, h\}\)
      2. \(E \cap F^C = \{t, a, i\}\)
      3. \((F \cap G)^C = S\)

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      MATH 140 WIR6 #4


  5. Let \(S = \{s_1, s_2, s_3, s_4, s_5\}\) be the sample space associated with the following partial probability distribution table.
    \[
    \begin{array}{l|c|c|c|c|c}
    \text{Outcome}   &  s_1 & s_2 & s_3 & s_4 & s_5\\
    \hline
    \text{Probability}  & \dfrac{3}{20} & \dfrac{7}{20} & \quad & \dfrac{4}{20} & \dfrac{1}{20}
    \end{array}
    \]Let \(E = \{s_3, s_4\}, F=\{ s_4, s_5\}\), and \(G=\{s_1, s_2, s_3\}\), find:
    1. \(P(E)\)
    2. \(P(F^C)\)
    3. \(P(E \cap G)\)
    4. \(P(G \cup E)\)
    5. \(P(F \cap G)\)​

      1. \(\dfrac{9}{20}\)
      2. \(\dfrac{15}{20} \)
      3. \(\dfrac{5}{20}\)
      4. \(\dfrac{19}{20}\)
      5. \(0\)

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      MATH 140 WIR6 #5


  6. An experiment consists of rolling two distinguishable four-sided die and observing the two numbers that land uppermost.
    1. Let \(Y\) be the random variable denoting the number of fours that land uppermost on the dice. Create the probability distribution for \(Y\).
    2. Let \(X\) be the random variable denoting the product of the two numbers. Create the
      probability distribution for \(X\) and then find the expected value for \(X\).​

      1. \(\begin{array}{c|c c c}
                Y & 0 & 1 & 2 \\[3pt]
                \hline
                P(Y) & \dfrac{9}{16} & \dfrac{6}{16} & \dfrac{1}{16}\\[5pt]
            \end{array}\)

      2. \(\begin{array}{c|c c c c c c c c c}
                X & 1 & 2 & 3 & 4 & 6 & 8 & 9 & 12 & 16 \\[5pt]
                \hline
                P(X) & \dfrac{1}{16} & \dfrac{2}{16} & \dfrac{2}{16} & \dfrac{3}{16} & \dfrac{2}{16} & \dfrac{2}{16} & \dfrac{1}{16} & \dfrac{2}{16} & \dfrac{1}{16}\\[5pt]
            \end{array}\)

        \(E(X) = 6.25\)

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      MATH 140 WIR6 #6


  7. Use the given partial histogram below, to answer the questions that follow.
    140_WIR6_Fig1.png
    1. Compute the value of \(P(X = 4.)\)
    2. Compute the value of \(P(X \leq 2 ).\)
    3. Compute the value of \(P(-2 < X \leq 4).\)
    4. Compute the value of \(P(X \geq 0).\)
    5. Calculate \(E(X).\)​

      1. \(P(X = 4) = 0.2\)
      2. \(P(X \leq 2 ) = 0.45\)
      3. \(P(-2 < X \leq 4) = 0.3 + 0.1 + 0.2 = 0.6\)
      4. \(P(X \geq 0) = 1 - 0.05 = 0.95\)
      5. \(E(X) = -2(0.05) + 0(0.3)+ 2(0.1) + 4(0.2) + 6(0.35) = 3\)

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      MATH 140 WIR6 #7


  8. For the linear programming problem below, define the variables, set up the objective function and constraints, then determine if it is a standard maximization problem.
    A company manufactures Products A, B, and C. Each product is processed in three departments: Dept.
    I, Dept. II, and Dept. III. The total available labor-hours, per week, for departments I, II, and III are
    800 hours, 1080 hours, and 840 hours, respectively. The number of labor-hours required to manufacture each product in each department, as well as the  profit realized per product for each of the three products are displayed in the table below.
      Product A Product B Product C
    Dept. I 3 hours 1 hour 3 hours
    Dept. II 4 hours 2 hours 2 hours
    Dept. III 2 hours 3 hours 1 hour
    Profit $16 $20 $15
    How many units of each product should the company produce each week to maximize their profit? ​

    1. Let x by the number of Product A manufactured each week.
      Let y by the number of Product B manufactured each week.
      Let z by the number of Product C manufactured each week.
    2. Objective function: \(P = 16x + 20y + 15z\)
              Constraints: \[
              \begin{cases}
              3x + y + z \leq 800 \\
              4x + 2y + 2z \leq 1080 \\
              2x + 3y + z \leq 840 \\
              x, y, z \geq 0
              \end{cases}
              \]
    3. Yes, standard max.

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    MATH 140 WIR6 #9