Homework #3

EE 364: Spring 2026

Assignment Details

Assigned: 27 January Due: Tuesday, 03 February at 16:00

BrightSpace Assignment: Homework 3

Instructions

Write your solutions to these homework problems. Submit your work to BrightSpace by the due date. Show all work and box answers where appropriate. Do not guess.

Problem 0

Daily derivation #3, Prove the Binomial Theorem.

Problem 1

A hockey team has 6 forwards, 4 defensemen, and 2 goalies. At any time, 3 forwards, 2 defensemen, and 1 goalie can be on the ice. How many combinations of players can a coach put on the ice?

Problem 2

Find the probability that in a class of 28 students exactly 4 were born on each of the seven days of the week.

Problem 3

A closet contains 10 pairs of shoes. You select 8 shoes at random. What is the probability that you select: (a) no complete pair, (b) exactly one complete pair, and (c) exactly two complete pairs?

Problem 4

$N$ students attend a Rush Week party. Each tosses their coat by the door. Each person grabs a coat uniformly at random from the pile when they leave. What is the probability that at least one student gets their correct coat?

Problem 5

Consider a group of $n$ students that includes Alice and Bob.

Suppose that the students stand in a row. What is the probability that there are exactly d students between Alice and Bob? Assume $d \le n - 2$.
Suppose instead that the students stand in a circle. Show that the probability that there are exactly $d$ students between Alice and Bob is $1/(n-1)$ and hence does not depend on $d$. Count only the students between Alice to Bob in a clockwise direction.

Problem 6

Prove the identity: \[\sum_{k=0}^{n} \frac{1}{k + 1} {n \choose k} = \frac{1}{n+1} \left(2^{n+1} - 1\right).\]

Problem 7

Two bags of apples sit before you on a table. The first bag contains two green apples and four red apples. The second bag contains five green apples and five red apples. Flip a fair coin to decide which bag to choose. Then pick four apples with replacement from that bag: pick an apple at random and then put it back in the bag and repeat three more times. Is it more likely than not that you will have picked two green apples and two red apples?

Problem 8

You have an unfair die that comes up “1” with probability 0.5 and all other outcomes have equal probability. Suppose that you place the unfair die in a box with 7 fair dice. You then randomly select one die and roll it six times. You do not observe any 2’s in the six rolls. Does the evidence increase or decrease your estimate of the probability that you selected the unfair die?

Problem 9

Alice and Bob play a coin flipping game. Bob flips a fair coin $n + 1$ times and Alice flips a coin $n$ times. Bob wins the game if he receives more heads than Alice. Else Alice wins. Is this a fair game?

Problem 10

A student body elects twelve students to the student government council. It turns out that four of the students are Democrats and four are Republicans with the rest undecided. The council’s first order of business is that they must choose four members from among themselves to serve on the executive committee. But the students cannot agree on whom to choose because both the Democrat and Republican students fear that students from the other party will dominate the committee. One of the undecided students gets tired of the bickering and says that fairness demands that they pick the four members at random. The student further states that it is likely that such a randomly chosen committee will contain an equal number of Democrats and Republicans. Do you agree?

Problem 11

John performs infrared inspections of high-voltage transformers for the Los Angeles Department of Water and Power (LADWP). John uses a thermal imaging scanner to inspect twelve transformers. He finds that three of the transformers are not-working and nine of the transformers are working. John sets reports for the twelve transformers on his desk. A gust of wind blows reports for three transformers to the ground. What is the probability that at least two of these reports are for working transformers?

Problem 12

John takes an intern position for a small company that makes flat screens for imported personal computers. His first assignment is to estimate the probability of rejecting a randomly sampled box of 10 flat screens. But his boss explains that there are two complications. The first is that company policy demands that he inspect exactly three screens per box of 10 and then that he accept the box only if all three screens are not defective. The second is that 70% of the boxes have only one defective screen in them while 30% of the boxes have four defective screens in them. John’s boss wants to know if this means that more than half the boxes will be rejected. How should John answer?

Problem 13

John goes to a meeting of the California Rare Fruit Growers Association at Cal State Fullerton. He walks past table after table of seeds and cuttings from many exotic plant and tree cultivars. The rarer cultivars are quite expensive. John stops at the pomegranate table. He samples watered-down juice from some rare pomegranate varieties from Armenia and decides he would like to grow some of these pomegranates. He has just enough money to buy a few viable seeds from three of the Armenian varieties. So he buys three seeds of a red pomegranate and five seeds of an orange pomegranate. He also buys seven seeds of the extra-rare white pomegranate. John puts the indistinguishable seeds in his pants pocket where before long they mix into a single small clump of seeds. That night John reaches into his pocket and pulls out four seeds and gives them to a friend to plant. John assures his friend that it is likely that at least one of the seeds is a white pomegranate. Do you agree?

Problem 14

Suppose that there are $n$ discrete probability density functions $f_1, f_2, \ldots, f_n$, defined on the same sample space $\Omega$. A probability density is a nonnegative function on the elements $k \in \Omega$ that sums to one. Suppose further that the $n$ nonnegative weights $w_1, w_2, \ldots, w_n$ also sum to one. Form the weighted mixture $f$ as follows: \[f(k) = \sum_{j=1}^{n} w_j f_j(k).\] Is the mixture $f$ a probability density function?

--- title: "Homework #3" subtitle: "EE 364: Spring 2026" sidebar: main format: html: toc: false number-sections: false code-fold: show code-tools: true theme: cosmo --- ::: {.callout-important} ## Assignment Details **Assigned:** 27 January **Due:** Tuesday, 03 February at 16:00 **BrightSpace Assignment:** [Homework 3](https://brightspace.usc.edu) ::: ::: {.callout-warning} ## Instructions Write your solutions to these homework problems. Submit your work to BrightSpace by the due date. Show all work and box answers where appropriate. Do not guess. ::: --- ## Problem 0 **Daily derivation #3**, Prove the Binomial Theorem. ## Problem 1 A hockey team has 6 forwards, 4 defensemen, and 2 goalies. At any time, 3 forwards, 2 defensemen, and 1 goalie can be on the ice. How many combinations of players can a coach put on the ice? ## Problem 2 Find the probability that in a class of 28 students exactly 4 were born on each of the seven days of the week. ## Problem 3 A closet contains 10 pairs of shoes. You select 8 shoes at random. What is the probability that you select: (a) no complete pair, (b) exactly one complete pair, and (c) exactly two complete pairs? ## Problem 4 $N$ students attend a Rush Week party. Each tosses their coat by the door. Each person grabs a coat uniformly at random from the pile when they leave. What is the probability that at least one student gets their correct coat? ## Problem 5 Consider a group of $n$ students that includes Alice and Bob. a. Suppose that the students stand in a row. What is the probability that there are exactly d students between Alice and Bob? Assume $d \le n - 2$. b. Suppose instead that the students stand in a circle. Show that the probability that there are exactly $d$ students between Alice and Bob is $1/(n-1)$ and hence does not depend on $d$. Count only the students between Alice to Bob in a clockwise direction. ## Problem 6 Prove the identity: $$\sum_{k=0}^{n} \frac{1}{k + 1} {n \choose k} = \frac{1}{n+1} \left(2^{n+1} - 1\right).$$ ## Problem 7 Two bags of apples sit before you on a table. The first bag contains two green apples and four red apples. The second bag contains five green apples and five red apples. Flip a fair coin to decide which bag to choose. Then pick four apples with replacement from that bag: pick an apple at random and then put it back in the bag and repeat three more times. Is it more likely than not that you will have picked two green apples and two red apples? ## Problem 8 You have an unfair die that comes up "1" with probability 0.5 and all other outcomes have equal probability. Suppose that you place the unfair die in a box with 7 fair dice. You then randomly select one die and roll it six times. You do not observe any 2's in the six rolls. Does the evidence increase or decrease your estimate of the probability that you selected the unfair die? ## Problem 9 Alice and Bob play a coin flipping game. Bob flips a fair coin $n + 1$ times and Alice flips a coin $n$ times. Bob wins the game if he receives more heads than Alice. Else Alice wins. Is this a fair game? ## Problem 10 A student body elects twelve students to the student government council. It turns out that four of the students are Democrats and four are Republicans with the rest undecided. The council's first order of business is that they must choose four members from among themselves to serve on the executive committee. But the students cannot agree on whom to choose because both the Democrat and Republican students fear that students from the other party will dominate the committee. One of the undecided students gets tired of the bickering and says that fairness demands that they pick the four members at random. The student further states that it is likely that such a randomly chosen committee will contain an equal number of Democrats and Republicans. Do you agree? ## Problem 11 John performs infrared inspections of high-voltage transformers for the Los Angeles Department of Water and Power (LADWP). John uses a thermal imaging scanner to inspect twelve transformers. He finds that three of the transformers are not-working and nine of the transformers are working. John sets reports for the twelve transformers on his desk. A gust of wind blows reports for three transformers to the ground. What is the probability that at least two of these reports are for working transformers? ## Problem 12 John takes an intern position for a small company that makes flat screens for imported personal computers. His first assignment is to estimate the probability of rejecting a randomly sampled box of 10 flat screens. But his boss explains that there are two complications. The first is that company policy demands that he inspect exactly three screens per box of 10 and then that he accept the box only if all three screens are not defective. The second is that 70% of the boxes have only one defective screen in them while 30% of the boxes have four defective screens in them. John's boss wants to know if this means that more than half the boxes will be rejected. How should John answer? ## Problem 13 John goes to a meeting of the California Rare Fruit Growers Association at Cal State Fullerton. He walks past table after table of seeds and cuttings from many exotic plant and tree cultivars. The rarer cultivars are quite expensive. John stops at the pomegranate table. He samples watered-down juice from some rare pomegranate varieties from Armenia and decides he would like to grow some of these pomegranates. He has just enough money to buy a few viable seeds from three of the Armenian varieties. So he buys three seeds of a red pomegranate and five seeds of an orange pomegranate. He also buys seven seeds of the extra-rare white pomegranate. John puts the indistinguishable seeds in his pants pocket where before long they mix into a single small clump of seeds. That night John reaches into his pocket and pulls out four seeds and gives them to a friend to plant. John assures his friend that it is likely that at least one of the seeds is a white pomegranate. Do you agree? ## Problem 14 Suppose that there are $n$ discrete probability density functions $f_1, f_2, \ldots, f_n$, defined on the same sample space $\Omega$. A probability density is a nonnegative function on the elements $k \in \Omega$ that sums to one. Suppose further that the $n$ nonnegative weights $w_1, w_2, \ldots, w_n$ also sum to one. Form the weighted *mixture* $f$ as follows: $$f(k) = \sum_{j=1}^{n} w_j f_j(k).$$ Is the mixture $f$ a probability density function?