# Probability/ Stats Puzzles – 2 & 3 (Solutions)

If you’ve not seen/ attempted the puzzles, the links are here: puzzle-2 and puzzle-3. These were presented in earlier posts.

Both these puzzles are adopted from a delightful little book by John Allen Paulos titled A Mathematician Reads the Newspaper.

I will provide more details about the book next week. For now, here are the solutions to the two puzzles.

## Puzzle-2

You need to call the throw of a dice a 1000 times. Like all dices, in each throw, this dice also gives you a number between 1 and 6. You are also told that the dice is slightly distorted / damaged – the probability of getting the six results is as follows: 1- 20%; 2- 10%; 3- 25%; 4-15%; 5-15%; 6-15%.

What strategy would you use to call the answers for the 1000 throws? Your objective is to get the right answer for a maximum of the throws.

Solution:

Call 3, 3, 3, 3…. all the 1000 times. This will get you aprroximately 250 right calls.

Or better still, tell the dice roller that your call is 3 all the thousand times, go for a coffee, or do something useful, come back after some time.

## Puzzle-3

Two contestants are to decide on the winner of 10 mn by flipping a coin. The winner will be the one who reaches six (6) correct calls first.

After 8 flips, contestant A has 5 correct calls, and contestant B has 3 correct calls. At this stage they agree NOT to continue with the flipping of the coin. Here are some proposals on how the money should be shared:

1. Contestant A says that since he is leading, he should get the 10mn.
2. Contestant B says that since the flipping was called off before the final result, the 10mn should be shared equally.
3. The show-host says that TV quiz program sponsors should retain the 10mn, since both the contestants agreed to call off the contest.
4. Someone from the audience suggests that the prize money be split in the 5:3 ratio (5 for A and 3 for B), in line with the number of right calls
5. A mathematician calls in to suggest that the money be split A7:B1 (try and guess the logic here, it is related to the probability of winning from this point, if the flipping had continued)

Solution:

The question on how the money is to be shared is not a mathematical /statistical problem at all! It is a matter of fairness and justice, and each solution proposed (and some yet to be proposed) has its own merit.

However, if you have not yet worked out the logic of why the mathematician proposed option # 5 above, here it is:

For contestant B to win 6 calls in a row, he/ she needs to call ALL of the next three calls correctly (even if he / she calls one incorrectly, A will reach 6 right calls. So the probability of B winning is (0.5) x (0.5) x (0.5) = 0.125; which means A has a probability of 0.875 – that is 7:1.

Next week, I will cover the source of these puzzles, a book titled A Mathematician Reads the Newspaper by John Allen Paulos.

You may also forward the link to this post to your friends, colleagues, and anyone else who may be interested.

Notes:

Nothing Official About It! – The views presented above are in no manner reflective of the official views of any organization, community, group, institute, country, government, or association. They may not even be the official views of the author of this post :-).

# Probability/ Stats Puzzle – 3

I encountered another problem in the same book (I will disclose the name of the book in a later post along with the answer). Here is the problem:

Two contestants have reached the last round of a TV quiz contest and one of them is hoping to be the winner of a prize of 10 mn (currency deliberately left vague) via a tie-breaker. Even after the tie-breaker, neither of them has beaten the other.

The show-host offers to break the tie with a coin (my guess is that the show host did not have any more questions left :-)). However, to maintain the suspense and gain more TRP, he proposes that the winner will be one who reaches six (6) correct calls first.

After 8 flips, contestant A has 5 correct calls, and contestant B has 3 correct calls. At this stage both the contestants agree NOT to continue with the flipping of the coin (maybe the coin is lost or it breaks or falls into something disgusting – use your imagination). They have to decide on the winner based on result of the 8 flips.

Here are some proposals:

1. Contestant A says that since he is leading, he should get the 10mn.
2. Contestant B says that since the flipping was called off before the final result, the 10mn should be shared equally.
3. The show-host says that TV quiz program sponsors should retain the 10mn, since both the contestants agreed to call off the contest.
4. Someone from the audience suggests that the prize money be split in the 5:3 ratio (5 for A and 3 for B), in line with the number of right calls
5. A mathematician calls in to suggest that the money be split A7:B1 (try and guess the logic here, it is related to the probability of winning from this point, if the flipping had continued)
6. Any other…

It is interesting to note so many options to a simple situation.

# Probability/ Stats Puzzle – 2

I encountered this simple problem in a book (I will disclose the name of the book in a later post along with the answer).

Here is the problem:

You need to call the throw of a dice a 1000 times. Like all dices, in each throw, this dice also gives you a number between 1 and 6. You are also told that the dice is slightly distorted / damaged – the probability of getting the six results is as follows: 1- 20%; 2- 10%; 3- 25%; 4-15%; 5-15%; 6-15%.

What strategy would you use to call the answers for the 1000 throws? Your objective is to get the right answer for a maximum of the throws.

Here are some answers that I have heard:

1. Call the number ‘3’ all the 1000 times – this is the most common answer I have heard.
2. Call the numbers in the same pattern as the probability: 1- 200 times; 2- 100 times; 3- 250 times; 4-150 times; 5-150 times; 6-150 times.
3. Call the numbers randomly, ignoring the distortion in the dice.
4. A variation of 2 above is to call the numbers in the same pattern, but also taking into account the answers to the past throws, so that we try and keep the probabilities similar to the expected patterns. So if in the first 100 throws, 1 has already rolled more than 20% and 2 has been rolled less than 10%, then in the 101st throw, call 2 instead of 1, and so on.
5. There are other possible answers too – and the right one may not be listed above (this is not a mutiple choice question 🙂 )

Work out the reasons for your choice, not just make a choice. The reasons are more important.

This is a simple question, and you should get the right answer.

The answer will be posted later.

# Book Review – “Made To Stick” by Chip Heath and Dan Heath

The messages from this book have ‘stuck’ to me over seven years. I remember this book very well and have been implicitly and explicitly guided by concepts that I learnt when I read the book long ago.

The authors have managed to implement what they are trying to teach :-)!

The full title of the book Made to Stick: Why Some Ideas Take Hold and Others Come Unstuck – makes the intent of the authors very clear. It is about conveying ideas and messages more effectively to achieve whatever you are trying to achieve Anyway, here is a passage from the Introduction of the book:

“We wrote this book to help you make your ideas stick. By ‘stick’, we mean that your ideas are understood and remembered, and have a lasting impact – they change your audience’s opinions and behavior.”

— Introduction: What Sticks? – Made to Stick

## Details of Made to Stick

Made to Stick: Why Some Ideas Take Hold and Others Come Unstuck

Authors: Chip Heath and Dan Heath

Publishing Date: 2007

Publisher: Arrow Books

Formats Available In: Hardcover, Paperback, Kindle, Audio

Available at: Amazon.com, Amazon.in.
This book is an entertaining, practical guide to effective communication. Extremely well-written, it uses the principles that are proposed in the book for effectively making ideas stick with the audience. It continues with the idea of ‘stickiness’ earlier popularized by Malcolm Gladwell in his book The Tipping Point.

The book contains a number of examples, cases, urban legends, personal stories, and analysis that are used to support the principles of effective communication proposed by the authors. The “Clinic” in each Chapter is used to illustrate the application of the principles discussed to a specific case study or idea.

According to the book, there are six principles of effective communication that combine to form the acronym SUCCES (the last S from “success” is absent). These six principles are:

• Simplicity. Strip an idea to its core. Keep prioritizing the ideas till you have something simple and profound.
• Unexpectedness. Use surprise to grab the audience’s attention.
• Concreteness. Avoid the abstract. Avoid ambiguity. Use vivid images.
• Credibility. Use whatever will make people believe in the idea. This could be the ‘messenger’ or the way the message is conveyed. (“For instance, if you have the security contract for Fort Knox, you are in the running for any security contract“)
• Emotional. Form an association between something they do care about to connect to things they don’t yet care about. (The most frequent reason for unsuccessful advertising is advertisers who are so full of their own accomplishments – ‘the world’s best seed!’ – that they forget to tell us why we should buy it  – ‘to have the world’s best lawn!‘)
• Stories. People remember stories compared to abstract messages. (For example, Subway used the story of a man who lost a lot of weight while eating their diet sandwich instead of providing data on calories and fat content of the sandwich).

There are over a hundred examples of successful messaging to illustrate each of the six principles. The book is extremely easy to read and difficult to put down once you start.

I strongly recommend at least one read of the book to following professionals – they can keep the principles in mind while crafting messages and campaigns:

• Executive management
• Marketing/ sales folks
• HR Policy makers
• Consultants/ Trainers
• People trying to handle change management

The book is available in multiple formats at Amazon.com, Amazon.in.

Chip Heath is a Professor of Organizational Behavior in the Graduate School of Business at Stanford University. His research examines why certain ideas – ranging from urban legends to folk medical cures, from Chicken Soup for the Soul stories to business strategy myths – survive and prosper in the social marketplace of ideas.

Dan Heath is a Senior Fellow at Duke University’s CASE center, which supports entrepreneurs who are fighting for social good.

Chip and Dan Heath have jointly co-authored another book Switch: How to change things when change is hard.

You can also view this 4:44 min video where the authors Chip and Dan Heath are being interviewed (uploaded on youtube):

The book is available at: Amazon.com, Amazon.in.

## Quick Quiz

In the two you tube clips below, can you figure out which of the principles of SUCCES are most prominently used?

Clip-1: Series of “You Don’t Mess With Texas” ads used to address littering in the state of Texas with celebrities conveying the message