# 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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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
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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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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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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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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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. $$\begin{array}{c|c c c c c c c c c} X & 1 & 2 & 3 & 4 & 6 & 8 & 9 & 12 & 16 \\[5pt] \hline P(X) & \dfrac{1}{16} & \dfrac{2}{16} & \dfrac{2}{16} & \dfrac{3}{16} & \dfrac{2}{16} & \dfrac{2}{16} & \dfrac{1}{16} & \dfrac{2}{16} & \dfrac{1}{16}\\[5pt] \end{array}$$ $$E(X) = 6.25$$ To see the full video page and find related videos, click the following link. 7. Use the given partial histogram below, to answer the questions that follow. 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. $$E(X) = -2(0.05) + 0(0.3)+ 2(0.1) + 4(0.2) + 6(0.35) = 3$$ To see the full video page and find related videos, click the following link. 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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