# Homework

Homework is to be turned in at the beginning of the class.

No late homework will be accepted. However, your lowest homework score will be dropped.

Cooperation on homework is permitted (and encouraged), but if you work together, do not take any paper away with you--in other words, you can share your thoughts (say on a blackboard), but you have to walk away with only your understanding. In particular, you must write the solution up on your own. Please acknowledge any cooperative work at the end of each assignment.

Legend:

• M : David Morin, Probability: For the Enthusiastic Beginner.
Morin's set of exercises can be found here.
• GS : Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, second revised edition.
The whole book can be found here.

## Assignments

### Homework 0: Due Sep 12

During the x-hour we will be working on writing proofs. To get you ready for that, please:

1. Read: Ten Tips for Writing Mathematical Proofs (Katharine Ott)
2. Skim through: How to write proofs: a quick guide (Eugenia Cheng)

See Worksheet 1

### Homework 1: Due Sep 20

1. M: §8.1: # 8, 16
2. M: §8.2: # 1, 2
3. Use the Binomial Theorem to show that for any positive integer $n$ we have $\sum_{k = 0} ^{n} \binom{n}{k} = 2^n$.
4. Let $A$ and $B$ be events such that $P(A \cap B) = 1/6$, $P( \text{not } A) = 2/3$ and $P(B) = 1/4$. What is $P(A \cup B)$? What is $P(B \cap (\text{not } A))$?
5. There are 10 students in a class; Alice and Bob are best friends in the class. If the teacher lines them up at random, what is the probability that Alice and Bob will be standing next to each other in line?
6. A die is loaded in such a way that the probability of each face turning up is proportional to the number of dots on that face. (For example, a six is three times as probable as a two.) What is the probability of getting an even number in one throw?

See Worksheet 2

### Homework 2: Due Sep 27

1. Let $X$ be a random variable with distribution function $m_X(x)$ defined by $m_X(−1) = 1/5$, $m_X(0) = 1/5$, $m_X(1) = 2/5$, and $m_X(2) = 1/5$.
• (a) Let $Y$ be the random variable defined by the equation $Y = X + 3$. Find the distribution function of $Y$.
• (b) Let $Z$ be the random variable defined by the equation $Z = X^2$. Find the distribution function of $Z$.
2. If $P(A \cup B) = 11/12$ , $P(A) = 5/12$ and $P(\text{not }B) = 1/4$, find $P(A|B)$.
3. What can be concluded about the events $A$ and $B$ if $P(A|B) = P(B|A)$?
4. There are three coins in a box. One is a two-headed coin, another is a fair coin, and the third is a biased coin that flips heads 75 percent of the time. When one of the coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?
5. Prove that if $A$ and $B$ are independent, so are:
1. $A$ and $(\text{not }B)$
2. $(\text{not }A)$ and $(\text{not }B)$.
6. A die is rolled until the first time a six turns up. Let $X$ be the random variable that counts the total number of rolls (including the roll that is the six).
• (a) What is the sample space?
• (b) What is distribution function for $X$?
• (c) Find $P(X > 3)$.
7. Two cards are drawn successively from a deck of 52 cards. Find the probability that the second card is higher in rank than the first card. Hint: Show that $1 = P (\text{higher}) + P (\text{lower}) + P (\text{same})$ and use the fact that $P (\text{higher}) = P (\text{lower})$.
8. A company produces two different types of assortments of candy. Their two biggest sellers are “Equal Split” which is 50 percent Twix and 50 percent Kit-Kat (by quantity) and “Uneven Break” which is 30 percent Twix and 70 percent Kit-Kat. A store buys two boxes of Equal Split and one box of Uneven Break. Unfortunately, in transit the three boxes lost their labels. In an effort to determine which box is which, the delivery man chooses one of the three boxes at random and pulls from that box one piece of candy (at random).
• (a) What is the probability that the piece of candy chosen was a Kit-Kat?
• (b) If the piece of candy chosen was a Kit-Kat, what is the probability that the box chosen by the delivery man was Uneven Break?

### Homework 3: Due Oct 4

1. Buzz has three wooden boxes, and each box has two drawers, one on each side. Buzz takes three gold rings and three silver rings and distributes one into each drawer: one box gets gold rings in both drawers, one gets silver rings in both drawers, and one gets one gold and one silver. Now, Woody comes in the room and randomly picks one of the three boxes, and randomly opens one of the two drawers in the box. If that drawer contains a gold ring, then what is the probability that the other drawer also contains a gold ring?
2. Sue claims that she can distinguish between Pepsi and Coke 75 percent of the time. Mel bets that she cannot and is just guessing randomly. To settle this a bet is made: Sue is to be given ten small glasses, each having been filled with Pepsi or Coke, chosen by tossing a fair coin. Sue wins the bet if she gets seven or more correct. Find the probability that Mel wins if Sue has the ability that she claims. Find the probability that Mel wins if Sue is guessing
3. Four women, $A$, $B$, $C$, and $D$, check their hats, and the hats are returned in a random manner. Let $\Omega$ be the set of all possible permutations of $A, B, C, D$. Let $X_j$ = 1 if the $j$th woman gets her own hat back and 0 otherwise. What is the distribution of $X_j$? Are the $X_i$’s mutually independent?
4. A die is rolled twice. Let $X$ denote the sum of the two numbers that turn up, and $Y$ the difference of the numbers (specifically, the number on the first roll minus the number on the second). Show that $E(XY ) = E(X)E(Y )$. Are X and Y independent?
5. You have 80 dollars and play the following game. A box contains two white balls and two black balls. You draw the balls out one at a time without replacement until all the balls are gone. On each draw, you bet half of your present fortune that you will draw a white ball. What is your expected final fortune?
6. Someone offers you the following game: You roll a fair six-sided die. If you roll a 1, you win $\$25$. If you roll a 2, you win$\$5$. If you roll a 3, you win nothing. If you roll a 4 or a 5, you lose $\$10$. If you roll a 6, you lose$\$15$. Should you play this game?

### Homework 4: Due Oct 11

1. A number is chosen at random from the set $S = \{−1,0,1\}$. Let $X$ be the number chosen. Find the expected value, variance, and standard deviation of $X$.
2. Suppose we have a box containing $c$ yellow balls and $d$ green balls. We draw $k$ balls, without replacement, from the box. Find the expected number of yellow balls drawn. Hint: Write the number of yellow balls drawn as the sum of $k$ random variables.
3. X is a random variable with $E(X) = 100$ and $V (X) = 15$. Find $E(X^2)$, $E(3X + 10)$, $E(-X)$ and $V(-X)$.
4. Suppose that $n$ people have their hats returned at random. Let $X_i = 1$ if the $i$th person gets his or her own hat back and $0$ otherwise. Let $S_n = \sum_{i = 1} ^n X_i$. Then $S_n$ is the total number of people who get their own hats back. Show that
1. $E(X_i^2) = 1/n.$
2. $E(X_i \cdot X_j) = 1/( n (n-1))$ for $i \neq j$.
3. $E(S_n ^2) = 2$.
4. $V(S_n) = 1$.
5. Suppose you choose at random a real number $X$ from the interval $[2, 10].$
1. Find the density function $f(x)$ and the probability of an event $E$ for this experiment, where E is a subinterval $[a, b]$ of $[2, 10]$.
2. Find the probability that $X > 5$.
3. Find the probability that $5 < X < 7$.
4. Find the probability that $X^2 − 12X + 35 > 0.$
6. A radioactive material emits $\alpha$-particles at a rate described by the density function $f(t) = .1e^{−.1t}$. Find the probability that a particle is emitted in the first 10 seconds, given that
1. no particle is emitted in the first second.
2. no particle is emitted in the first 5 seconds.
3. a particle is emitted in the first 3 seconds.
4. a particle is emitted in the first 20 seconds.

### Homework 5: Due Oct 18

1. On the average, only 1 person in 1000 has a particular rare blood type.
1. Find the probability that, in a city of 10,000 people, no one has this blood type.
2. How many people would have to be tested to give a probability greater than 1/2 of finding at least one person with this blood type?
2. An advertiser drops 10,000 leaflets on a city which has 2000 blocks. Assume that each leaflet has an equal chance of landing on each block. What is the probability that a particular block will receive no leaflets?
3. An airline finds that 4 percent of the passengers that make reservations on a particular flight will not show up. Consequently, their policy is to sell 100 reserved seats on a plane that has only 98 seats. Find the probability that every person who shows up for the flight will find a seat available.
4. On an average 8-hour school day, 1000 people walk into Kemeny Hall. Assume this happens completely randomly. What is the probability that exactly six people enter Kemeny Hall in a given ten minute span?

### Homework 6: Due Oct 25

1. Let $X_1, X_2$ be independent, uniform random variables on $\{1, 2, \cdots , 10\}$. Find $$P(\text{the maximum of X_1, X_2 is at least 5}).$$
2. Let $X$ and $Y$ be independent random variables with uniform density functions on $[0,1]$. Find
1. $E(|X - Y|)$.
2. $E(\max(X,Y))$.
3. $E(\min(X,Y))$.
4. $E(X^2 + Y^2)$.
5. $E((X + Y)^2)$.
3. Bridies' Bearing Works manufactures bearing shafts whose diameters are normally distributed with parameters $\mu = 1$, $\sigma = .002$. The buyer's specifications require these diameters to be $1.000 \pm .003$ cm. What fraction of the manufacturer's shafts are likely to be rejected? If the manufacturer improves her quality control, she can reduce the value of $\sigma$. What value of $\sigma$ will ensure that no more than 1 percent of her shafts are likely to be rejected?
4. A fair coin is tossed 100 times. The expected number of heads is 50, and the standard deviation for the number of heads is 5 (check this). What does Chebyshev’s Inequality tell you about the probability that the number of heads that turn up deviates from the expected number 50 by three or more standard deviations (i.e., by at least 15)?
5. A student's score on a particular calculus final is a random variable with values of $[0,100]$, mean 70, and variance 25.
1. Find a lower bound for the probability that the student's score will fall between 65 and 75.
2. If 100 students take the final, find a lower bound for the probability that the class average will fall between 65 and 75.
6. A share of common stock in the Pilsdorff beer company has price $Y_n$ on the $n$th business day of the year. ($Y_n$ is a random variable.) Finn observes that the price change $X_n = Y_{n+1} - Y_n$ appears to be a random variable with mean $\mu = 0$ and variance $\sigma^2 = 1/4$. If $Y_1 = 30$, find a lower bound for the following probabilities, under the assumption that the $X_n$'s are mutually independent.
1. $P(25 \leq Y_2 \leq 35)$
2. $P(25 \leq Y_{11} \leq 35)$
3. $P(25 \leq Y_{101} \leq 35)$

See Worksheet 6

### Homework 7: Due Nov 1

1. Let $S_{100}$ be the number of heads that turn up in 100 tosses of a fair coin. Use the Central Limit Theorem to estimate
1. $P(S_{100} \leq 45)$.
2. $P(45 < S_{100} < 55)$.
3. $P(S_{100} > 63)$.
4. $P(S_{100} < 57)$.
2. Let $S_n$ be the number of successes in $n$ Bernoulli trials with probability .8 for success on each trial. Let $A_n = S_n/n$ be the average number of successes. In each case give the value for the limit, and give a reason for your answer.
1. $\lim_{n \to \infty} P(A_n = .8)$.
2. $\lim_{n \to \infty} P(.7n < S_n < .9n)$.
3. $\lim_{n \to \infty} P(S_n < .8n + .8\sqrt n)$.
4. $\lim_{n \to \infty} P(.79 < A_n < .81)$
3. Let $S$ be the number of heads in 1,000,000 tosses of a fair coin. Use (a) Chebyshev's inequality, and (b) the Central Limit Theorem, to estimate the probability that $S$ lies between 499,500 and 500,500. Use the same two methods to estimate the probability that $S$ lies between 499,000 and 501,000, and the probability that $S$ lies between 498,500 and 501,500.
4. A noodle machine in Spumoni's spaghetti factory makes about 5 percent defective noodles even when properly adjusted. The noodles are then packed in crates containing 1900 noodles each. A crate is examined and found to contain 115 defective noodles. What is the approximate probability of finding at least this many defective noodles if the machine is properly adjusted?
5. In an opinion poll it is assumed that an unknown proportion p of the people are in favor of a proposed new law and a proportion 1 − p are against it. A sample of n people is taken to obtain their opinion. The proportion p ̄ in favor in the sample is taken as an estimate of p. Use (a) Chebyshev's inequality, and (b) the Central Limit Theorem, to determine how large a sample will ensure that the estimate will, with probability .95, be correct to within .01.
6. A surveying instrument makes an error of $-2$,$-1$, 0, 1, or 2 feet with equal probabilities when measuring the height of a 200-foot tower.
1. Find the expected value and the variance for the height obtained using this instrument once.
2. Estimate the probability that in 18 independent measurements of this tower, the average of the measurements is between 199 and 201, inclusive.
7. A share of common stock in the Pilsdorff beer company has price $Y_n$ on the $n$th business day of the year. ($Y_n$ is a random variable.) Finn observes that the price change $X_n = Y_{n+1} - Y_n$ appears to be a random variable with mean $\mu = 0$ and variance $\sigma^2 = 1/4$. If $Y_1 = 30$, use the Central Limit Theorem to estimate the following probabilities, under the assumption that the $X_n$'s are mutually independent.
1. $P(25 \leq Y_2 \leq 35)$
2. $P(25 \leq Y_{11} \leq 35)$
3. $P(25 \leq Y_{101} \leq 35)$

See Worksheet 7

### Homework 8: Due Nov 8

GS:§11.1: #2, 3, 4, 6, 7, 19
GS: §11.2: # 1,2,3