DASC 6000: Written Assignment 02
Assignment Goal
The overarching goal for this assignment is to assess your understanding of sample spaces and probability.
1 De Morgan’s Laws
Consider rolling a six-sided die. Let \mathbb{A} be the set of outcomes where the roll is an even number. Let \mathbb{B} be the set of outcomes where the roll is greater than 3. Calculate and compare the sets on both sides of the following De Morgan’s laws:
(\mathbb{A} \cup \mathbb{B})^c=\mathbb{A}^c \cap \mathbb{B}^c
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(\mathbb{A} \cap \mathbb{B})^c = \mathbb{A}^c \cup \mathbb{B}^c
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2 Set Identities
Let \mathbb{A} and \mathbb{B} be two sets. Show that:
\mathbb{A}^c=\left(\mathbb{A}^c \cap \mathbb{B}\right) \cup\left(\mathbb{A}^c \cap \mathbb{B}^c\right)
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\mathbb{B}^c=\left(\mathbb{A} \cap \mathbb{B}^c\right) \cup\left(\mathbb{A}^c \cap \mathbb{B}^c\right)
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(\mathbb{A} \cap \mathbb{B})^c=\left(\mathbb{A}^c \cap \mathbb{B}\right) \cup\left(\mathbb{A}^c \cap \mathbb{B}^c\right) \cup\left(\mathbb{A} \cap \mathbb{B}^c\right)
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Consider rolling a fair six-sided die. Let \mathbb{A} be the set of outcomes where the roll is an odd number. Let \mathbb{B} be the set of outcomes where the roll is less than 4. Calculate the sets on both sides of the equality in part (3), and verify that the equality holds.
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3 Principle of Inclusion and Exclusion
Out of the students in a class, 60 \% are geniuses, 70 \% love chocolate, and 40 \% fall into both categories. Determine the probability that a randomly selected student is neither a genius nor a chocolate lover.
Your answer goes here.4 Constructing a Probabilistic Model
A six-sided die is loaded in a way that each even face is twice as likely as each odd face. All even faces are equally likely, as are all odd faces. Construct a probabilistic model for a single roll of this die and find the probability that the outcome is less than 4.
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5 Sample Space
A four-sided die is rolled repeatedly, until the first time (if ever) that an even number is obtained. What is the sample space for this experiment?
Your answer goes here.6 Sample Space Partitioning
A partition of the sample space \Omega is a collection of disjoint events S_1, \ldots, S_n such that \Omega=\cup_{i=1}^n S_i
Show that for any event \mathbb{A}, we have \mathbf{P}(\mathbb{A})=\sum_{i=1}^n \mathbf{P}\left(\mathbb{A} \cap S_i\right)
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Use part (a) to show that for any events A, B, and C we have \mathbf{P}(\mathbb{A})=\mathbf{P}(\mathbb{A} \cap \mathbb{B})+\mathbf{P}(\mathbb{A} \cap \mathbb{C})+\mathbf{P}\left(\mathbb{A} \cap \mathbb{B}^c \cap \mathbb{C}^c\right)-\mathbf{P}(\mathbb{A} \cap \mathbb{B} \cap \mathbb{C})
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7 Six-sided Dice
We roll two fair six-sided dice. Each one of the 36 possible outcomes is assumed to be equally likely.
Find the probability that doubles are rolled.
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Given that the roll results in a sum of 4 or less, find the conditional probability that doubles are rolled.
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Find the probability that at least one die roll is a 6.
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Given that the two dice land on different numbers, find the conditional probability that at least one die roll is a 6.
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8 Fair and Unfair Coins
A coin is tossed twice. Alice claims that the event of two heads is at least as likely if we know that the first toss is a head than if we know that at least one of the tosses is a head. Is she right? Does it make a difference if the coin is fair or unfair? How can we generalize Alice’s reasoning?
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9 Biased Coins
We are given three coins: one has heads in both faces, the second has tails in both faces, and the third has a head in one face and a tail in the other. We choose a coin at random, toss it, and the result is heads. What is the probability that the opposite face is tails?
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10 Quality Assurance
A batch of one hundred items is inspected by testing four randomly selected items. If one of the four is defective, the batch is rejected. What is the probability that the batch is accepted if it contains five defectives?
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11 Q:
Alice searches for her term paper in her filing cabinet, which has several drawers. She knows that she left her term paper in drawer j with probability \mathbf{P}_j>O. The drawers are so messy that even if she correctly guesses that the term paper is in drawer i, the probability that she finds it is only d_i.
Alice searches in a particular drawer, say drawer i, but the search is unsuccessful. Conditioned on this event, show that the probability that her paper is in drawer j, is given by \begin{aligned} \frac{p_j}{1-p_i d_i} &, \quad \text { if } j \neq i \\ \\ \frac{p_i\left(1-d_i\right)}{1-p_i d_i}&, \quad \text { if } j=i \end{aligned} Your answer goes here.
12 Jars and Balls
Each of k jars contains m white and n black balls. A ball is randomly chosen from jar 1 and transferred to jar 2, then a ball is randomly chosen from jar 2 and transferred to jar 3, etc. Finally, a ball is randomly chosen from jar k. Show that the probability that the last ball is white is the same as the probability that the first ball is white, i.e., it is m /(m+n).
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13 Jars and Balls
We have two jars, each initially containing an equal number of balls. We perform four successive ball exchanges. In each exchange, we pick simultaneously and at random a ball from each jar and move it to the other jar. What is the probability that at the end of the four exchanges all the balls will be in the jar where they started?
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14 Coin Tossing
Alice and Bob have 2 n+1 coins, each coin with probability of heads equal to \frac{1}{2}. Bob tosses n+1 coins, while Alice tosses the remaining n coins. Assuming independent coin tosses, show that the probability that after all coins have been tossed, Bob will have gotten more heads than Alice is \frac{1}{2}.
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15 Message Transmission Over a Communications Channel
A source transmits a message (a string of symbols) through a noisy communication channel. Each symbol is 0 or 1 with probability p and 1-p, respectively, and is received incorrectly probability \epsilon_0 and \epsilon_1, respectively (see the figure below). Errors in different symbol transactions are independent.
What is the probability that k th symbol is received correctly?
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What is the probability that the string of 1011 is received correctly?
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In an effort to improve reliability, each symbol is transmitted three time and received string is decoded by majority rule. In other words a 0 (or 1 ) is transmitted as 000 (or 111, respectively), and it is decoded at the receiver as a 0 (or 1) if and only if the received three-symbol string contains at least two 0s (or 1s, respectively). What is the probability that a 0 is correctly decoded?
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For what values of \epsilon_0 is there an improvement in the probability of correct decoding of a 0 when the scheme of part (c) is used?
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Suppose that the scheme of part (c) is used. What is the probability that a symbol was 0 given that the received string is 101?
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16 King’s Siblings
The king has only one sibling. What is the probability that the sibling is male? Assume that every birth results in a boy with probability \frac{1}{2} independent of other births. Be careful to state any additional assumptions you have to make in order to arrive at an answer.
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17 Fair and Biased Coins
Alice and Bob want to choose between the opera and the movies by tossing a fair coin. Unfortunately, the only available coin is biased (though the bias is not known exactly). How can they use the biased coin to make a decision so that either option (opera or the movies) is equally likely to be chosen?
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18 Probability of a System Outage
A cellular phone system services a population of n_1 voice users (those who occasionally need a voice connection) and n_2 data users (those who occasionally need a data connection). We estimate that at a given time, each user will need to be connected to the system with probability p_1 (for voice users) or p_2 (for data users), independent of other users. The data rate for a voice user is r_1 bits/sec and for a data user is r_2 bits/sec. The cellular system has a total capacity of c bits/sec. What is the probability that more users want to use the system than the system can accommodate?
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19 Coin Tossing
Consider a coin that comes up heads with probability p and tails with probability 1-p. Let q_n be the probability that after n independent tosses, there have been an even number of heads. Derive a recursion that relates q_n to q_n-1, and solve this recursion to establish the formula: q_n=\left(1+(1-2 p)^n\right) / 2 Your answer goes here.
20 Probability of Events
A six-sided die is rolled three times independently. Which is more likely: a sum of 11 or a sum of 12?
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21 Birhday Problem
Consider n people who are attending a party. We assume that every person has an equal probability of being born on any day during the year, independent of everyone else, and ignore the additional complication presented by leap years (i.e., assume that nobody is born on February 29). What is the probability that each person has a distinct birthday?
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22 Urn Model
An urn contains m red and n white balls.
We draw two balls randomly and simultaneously. Describe the sample space and calculate the probability that the selected balls are of different color, by using two approaches: a counting approach based on the discrete uniform law, and a sequential approach based on the multiplication rule.
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We roll a fair 3-sided die whose faces are labeled 1, 2, 3, and if k comes up, we remove k balls from the urn at random and put them aside. Describe the sample space and calculate the probability that all of the balls drawn are red, using a divide-andconquer approach and the total probability theorem.
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23 Permutations and Combinations
An academic department offers 8 lower level courses: \left\{L_1, L_2, \ldots, L_8\right\} and to higher level courses: \left\{H_1, H_2, \ldots, H_{10}\right\}. A valid curriculum consists of 4 lower level courses and 3 higher level courses.
How many different curricula are possible?
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Suppose that \left\{H_1, \ldots, H_5\right\} have L_1 as a prerequisite, and \left\{H_6, \ldots H_{10}\right\} have L_2 and L_3 as prerequisites. i.e., any curricula which involve, say, one of \left\{H_1, \ldots, H_5\right\} must also include L_1. How many different curricula are there?
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24 Computing Probability
We draw the top 7 cards from a well-shuffled standard 52-card deck. Find the probability that:
The 7 cards include exactly 3 aces.
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The 7 cards include exactly 2 kings.
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The probability that the 7 cards include exactly 3 aces or exactly 2 kings, or both.
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