## Thursday, December 5, 2013

### A Cow, A Monkey and a Tree

Q: A Cow is tethered to a tree by a rope that has half the length of the circumference of the tree's stump. A monkey with a stone in hand jumps onto the tree and starts hopping around the tree's branches which covers a circular area of radius five times that of the stump. This causes the cow to run about at random and monkey drops the stone at random. What is the probability that the cow would get hit?

A: The situation can be characterized by the figure shown below (not to scale)

The monkey can be anywhere in the light shaded orange area before it drops the stone (this of course assumes that the stump has no terrace area). The cow, can sweep a spiral as shown in the figure below along either lines (poorly drawn).

To start, lets first compute the area the cow can sweep. Assume the cow moves a bit to the left, the rope would wind up along the stump of the tree. To further formalize this assume it sweeps out an angle $$\theta$$ at the center as shown figure below.
In the figure if the cow's rope winds up by a length AB subtending an angle $$\theta$$ at the center C, and if the cow moves from D to E an incremental angle of $$d\theta$$ is subtended at the centre. The length of the rope is now decreased from $$\pi r$$ to $$\pi r - r\theta$$. The length of DE can be computed as
$$(\pi r - r\theta )d\theta$$
The incremental area of triangle ADE is $$\frac{1}{2}\times DE \times AD$$ which works out as
$$dA = \frac{1}{2}(\pi r - r \theta)^{2}d\theta$$
To compute the total area, we can integrate out with $$\theta$$ running from $$[0,\pi]$$ and as the area is symmetric, we multiply by 2
$$A =2 \times \int_{0}^{\pi}\frac{1}{2}r^{2}(\pi - \theta)^{2}d\theta$$
The above integral simplifies to
$$A = \frac{r^{2}\pi^{3}}{3}$$
The total area the monkey can drop the stone is easy to compute as
$$A_{total} = \pi (5r)^{2} - \pi r^{2} = 24\pi r^{2}$$
The sought area and probability is simply the ratio of the two areas $$\frac{A}{A_{total}} = \frac{\pi^{3}}{24\times 3} \approx 43\%$$

Some of the best books to own to learn the art of probability

Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.

Introduction to Algorithms
This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists

Introduction to Probability Theory
Overall an excellent book to learn probability, well recommended for undergrads and graduate students

An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner.

Introduction to Probability, 2nd Edition
A good book to own. Does not require prior knowledge of other areas, but the book is a bit low on worked out examples.

Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)
An excellent resource (students, engineers and even entrepreneurs) if you are looking for some code that you can take and implement directly on the job

Understanding Probability: Chance Rules in Everyday Life
This is a great book to own. The second half of the book may require some knowledge of calculus. It appears to be the right mix for someone who wants to learn but doesn't want to be scared with the "lemmas"

Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems)
This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.

This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

This book has been yellow-flagged with some issues: including sequencing of content that could be an issue. But otherwise its good

## Thursday, November 28, 2013

### The Sultan's Wine Bottles

Q: A Sultan has a 1000 bottles of wine. He needs to use them in 30 days time for a royal banquet. He knows that his enemies have poisoned exactly one bottle with a type of poison that takes effect in 29 days. He decides to use his soldiers to test which bottle is poisoned. Is there a strategy that minimizes the number of soldiers needed for the task?

Probability Theory: The Logic of Science

A: The naive approach is to have one soldier per bottle. Every soldier gets a drop from each bottle and they wait for 29 days. The number of the soldier who gets affected on the 29th day shows which bottle is poisoned. However, this strategy is quite expensive in terms of the number of soldiers needed for the Sultan. A far more efficient strategy is the following.
1. Label each bottle with a number.
2. Maintain a ledger which maps a number to each of the patterns 0000000000 -> 1, 0000000001 -> 2, 0000000010 -> 3, 0000000100 -> 4 and so on till you reach 1111111111 -> 1000. Note there are 10 places that can hold either 0s or 1s.
3. Assign each soldier to each of the 10 places.
4. For each bottle number, give a drop of the wine to each soldier with a number 1.
What happens? On the 29th day, a certain combination of soldiers will be affected by the poison. Knowing that combination, the Sultan can trace back and find the exact bottle that contained the poison by looking up the ledger. This way the number of soldiers needed is minimized. Note, the problem solution can be extended to any number of bottles. By using the binary number encoding method the number of combinations needed can be ascertained by simply doing $$log_{2}{N}$$ where $$N$$ is the number of bottles.
If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.

Introduction to Algorithms
This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists

Introduction to Probability Theory
Overall an excellent book to learn probability, well recommended for undergrads and graduate students

An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner.

Introduction to Probability, 2nd Edition
A good book to own. Does not require prior knowledge of other areas, but the book is a bit low on worked out examples.

Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)
An excellent resource (students, engineers and even entrepreneurs) if you are looking for some code that you can take and implement directly on the job

Understanding Probability: Chance Rules in Everyday Life
This is a great book to own. The second half of the book may require some knowledge of calculus. It appears to be the right mix for someone who wants to learn but doesn't want to be scared with the "lemmas"

Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems)
This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.

This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

This book has been yellow-flagged with some issues: including sequencing of content that could be an issue. But otherwise its good

## Tuesday, November 26, 2013

### Drawing Increasing Numbers from a Deck of Cards

Q: You have a 5 decks of cards numbered 1-12. You draw 4 cards at random from each deck. What is the probability that the 4 cards come out in an increasing sequence from at least one of the draws?

Dremel CKDR-02 Ultimate 3-Tool Combo Kit with 15 Accessories and Storage Bag

A: Note a subtle point here. The number of cards in the deck to draw from doesn't really matter. All we are concerned with is if the numbers are in an increasing sequence. If you draw 4 cards from any sized deck, you are left with 4 distinct cards and that's it! Given 4 cards there are $$4! = 24$$ ways (permutations) to place them in any given order. Thus, the required scenario happens with a probability of $$\frac{1}{4!} = \frac{1}{24}$$.

Therefore,
1. The probability that they are not in an increasing sequence is $$1-\frac{1}{24}= \frac{23}{24}$$.
2. The probability that all 5 draws from each of the 5 decks is not in an increasing sequence is $$\big(\frac{23}{24})^{5}$$
3. The probability that at least 1 of the 5 draws from each of the 5 decks is in an increasing sequence $$= 1 - \big(\frac{23}{24})^{5}) = 19.16\%$$ which is the sought probability.
If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.

Introduction to Algorithms
This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists

Introduction to Probability Theory
Overall an excellent book to learn probability, well recommended for undergrads and graduate students

An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner.

Introduction to Probability, 2nd Edition
A good book to own. Does not require prior knowledge of other areas, but the book is a bit low on worked out examples.

Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)
An excellent resource (students, engineers and even entrepreneurs) if you are looking for some code that you can take and implement directly on the job

Understanding Probability: Chance Rules in Everyday Life
This is a great book to own. The second half of the book may require some knowledge of calculus. It appears to be the right mix for someone who wants to learn but doesn't want to be scared with the "lemmas"

Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems)
This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.

This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

This book has been yellow-flagged with some issues: including sequencing of content that could be an issue. But otherwise its good

## Thursday, November 21, 2013

### Winning the Lottery, Twice!

You might have heard this piece of news often "XYZ wins lottery for a second time". Such news, like here, leaves the reader wondering how someone can get that lucky. Needless to say, the winner is on cloud nine knowing s/he has won, not once but twice! But are these events truly rare? We are conditioned to believe that winning a lottery in itself is a rare event let alone winning it twice. Lets explore.

Probability Theory: The Logic of Science

To begin understanding the rarity of the above event, lets revisit the Birthday Problem. The summary of the problem is we need an astonishingly small number of people in a room to have $$>50\%$$ probability that two people would have the same birthday, about 23. To generalize, there are 365 days in a year. If there are 2 people in a room, the probability that both have different birthdays are $$\frac{364}{365}$$. If there are 3 its $$\frac{364}{365}\times\frac{363}{365}$$ and so on. If you assume there are $$N$$ days in a year and there are $$r$$ people we can compute the probability that they all have different birthdays as
$$P(\text{different birthday}) = \frac{N}{N}\times\frac{N-1}{N}\times\frac{N-2}{N}\ldots\frac{N-r-1}{N}$$
Implying the probability that at least two have the same birthday is $$1 - P(\text{different birthday})$$

These results can be extended to the double-win lottery phenomena as well. $$N$$ can be mapped to the possible numbers in a lottery, $$r$$ could be the number of lottery players. Let us make some simplifying assumptions.
• The winners of a lottery would continue buying tickets and most lottery buyers who don't win continue to buy lotteries, and assume all lotteries are drawn weekly.
• The lottery draw is done once a week.
• It is the same set of $$r$$ people who are buying every week and the winner is one of the $$r$$
• Everybody buys one ticket which has a distinct number.
To compute the probability that in the second week the same person does not win, two things need to happen.
1. The winning number has be one of the $$r$$, this happens with probability $$\frac{r}{N}$$
2. The winner in the first round should not be the winner in the second round. This happens with probability $$\frac{r-1}{r}$$.
Thus, the overall probability of not having a double winner in the second week is $$\frac{r}{N}\times\frac{r}{N}\times \frac{r-1}{r} = \frac{r(r-1)}{N^2}$$. For the 3rd week, by applying a similar logic, we get the probability that all three are distinct winners as

$$P(\text{all 3 distinct}) = \frac{r(r-1)(r-2)}{N^3}$$

Let us examine what happens over a decade. That is a total of 520 draws. Let us also put some real numbers behind this. Assume there are 10 million lottery players and there are 175 million numbers to choose from (these are rough estimates from Powerball numbers). The value of $$P(\text{all distinct})$$ works out as
$$\frac{10^{6}\times(10^{6} - 1)\times(10^{6} - 2)\ldots\times (10^{6} - 520)}{(175\times 10^{6})^{520}}$$
The above expression is in-computable by any machine we know of, however we can easily find a maximum possible value for this expression. The numerator is clearly less that $$10^{6\times 520}$$ and the denominator can be factored out as $$175^{6\times 520} \times 10^{6\times 520}$$. This gives a maximum bound for the expression as
$$\frac{1}{175^{520}} \approx 0$$
This implies, the probability of there being a double winner over a decade, under the prevailing assumptions, is almost $$100\%$$!
Why then have we not heard of double winners in Powerball jackpots? The most likely reason is the winners of big lottery draws likely don't bother coming back and buying more. Also, from the point of view of the buyer the probability of a double win IS astronomically small, however the probability that there would be a double winner somewhere or the other (under the prevailing assumptions) is quite the opposite, it is a mathematical certainty!

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.

Introduction to Algorithms
This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists

Introduction to Probability Theory
Overall an excellent book to learn probability, well recommended for undergrads and graduate students

An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner.

Introduction to Probability, 2nd Edition
A good book to own. Does not require prior knowledge of other areas, but the book is a bit low on worked out examples.

Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)
An excellent resource (students, engineers and even entrepreneurs) if you are looking for some code that you can take and implement directly on the job

Understanding Probability: Chance Rules in Everyday Life
This is a great book to own. The second half of the book may require some knowledge of calculus. It appears to be the right mix for someone who wants to learn but doesn't want to be scared with the "lemmas"

Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems)
This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.

This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

This book has been yellow-flagged with some issues: including sequencing of content that could be an issue. But otherwise its good

## Monday, November 4, 2013

### Points on a Circle

Q: A total of $$n$$ random points are selected from a circle. What is the probability that all of them are on a semicircle when evaluated in a clockwise order?

Probability Theory: The Logic of Science

A: Assume $$n=4$$ as shown in the figure below. Without loss of generality, assume the points were picked in the order shown by the numbers in the figure.
Start with the point numbered 1 and look in a clockwise direction. Each of the subsequent points can be either in or out of the semicircle starting from 1. Each of the given points will be within the semicircle with probability $$\frac{1}{2}$$. Therefore the probability that all the remaining $$n-1$$ points will be in the clockwise semicircle is $$\frac{1}{2^{n-1}}$$. This estimate is for a given starting point numbered at 1. To compute the probability for all $$N$$ points, we apply the same logic to each of the points in sequence. This yields $$\frac{n}{2^{n-1}}$$ which is the probability that all $$n$$ points lie on a semicircle. The same thinking can be extended for a quadrant of a circle too. For a quadrant, the probability that a given point lies within a quadrant is $$\frac{1}{4}$$ and the overall probability works out to $$\frac{n}{4^{n-1}}$$.

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.

Introduction to Algorithms
This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists

Introduction to Probability Theory
Overall an excellent book to learn probability, well recommended for undergrads and graduate students

An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner.

Introduction to Probability, 2nd Edition
A good book to own. Does not require prior knowledge of other areas, but the book is a bit low on worked out examples.

Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)
An excellent resource (students, engineers and even entrepreneurs) if you are looking for some code that you can take and implement directly on the job

Understanding Probability: Chance Rules in Everyday Life
This is a great book to own. The second half of the book may require some knowledge of calculus. It appears to be the right mix for someone who wants to learn but doesn't want to be scared with the "lemmas"

Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems)
This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.

This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

This book has been yellow-flagged with some issues: including sequencing of content that could be an issue. But otherwise its good

## Monday, October 28, 2013

### Lotto Urn with a Twist.

Q: An urn contains lotto tickets numbered 1 to 100. You play the game by paying a fee. The rules of the game are as follows, you need to draw two tickets, you are paid the value of the lower number of the tickets. What is a fair price to play this game?

Probability Theory: The Logic of Science

A: The first ticket drawn could be any of the tickets from 1 to 100. The second draw sets up the payoff which will be the lower of the two. Assume the first drawn ticket has the number $$i$$ on it. The second ticket drawn could be either greater or lesser with probabilitie as shown in the figure below.
If the chosen number is lesser, the expected value is $$\frac{i-1}{2}$$. Likewise, if the chosen value is greater the expected value is $$\frac{N-i}{2}$$. Thus, the total expected value can be computed as
$$E = \frac{i-1}{N}\times \frac{i-1}{2} + \frac{N-i}{N}\times\frac{N-i}{2}$$
Note, the above expression is just for the case when $$i^{th}$$ ticket is selected. This happens with probability $$\frac{1}{N}$$. So, the resulting expected value is
$$E = \frac{1}{N}\sum_{i=1}^{N}\frac{(i-1)^{2}}{2N} + \frac{(N-i)^{2}}{2N}$$
Using the identities for the sum of integers to $$n$$ and the sum of squares of integers to $$n^{2}$$, the above expression can be simplified to
$$\frac{n(n+1)(2n+1)}{3} - n(n+1)^{2} + n + n^{3}$$
Note the above expression is only valid for large $$N$$. As we have already selected one ticket there remains only $$N-1$$ tickets, but we can ignore this change for now.
Plugging in $$n = 100$$ yields the expected payoff to be $$\approx 32.5$$ which is the breakeven fee to play this game.

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.

Introduction to Algorithms
This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists

Introduction to Probability Theory
Overall an excellent book to learn probability, well recommended for undergrads and graduate students

An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner.

Introduction to Probability, 2nd Edition
A good book to own. Does not require prior knowledge of other areas, but the book is a bit low on worked out examples.

Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)
An excellent resource (students, engineers and even entrepreneurs) if you are looking for some code that you can take and implement directly on the job

Understanding Probability: Chance Rules in Everyday Life
This is a great book to own. The second half of the book may require some knowledge of calculus. It appears to be the right mix for someone who wants to learn but doesn't want to be scared with the "lemmas"

Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems)
This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.

This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

This book has been yellow-flagged with some issues: including sequencing of content that could be an issue. But otherwise its good