Test your intuition:

You throw a dice until you get 6. What is the expected number of throws (including the throw giving 6)** conditioned** on the event that all throws gave even numbers.

Test your intuition:

You throw a dice until you get 6. What is the expected number of throws (including the throw giving 6)** conditioned** on the event that all throws gave even numbers.

This entry was posted in Combinatorics, Probability, Test your intuition and tagged Elchanan Mossel, Test your intuition. Bookmark the permalink.

### Recent Comments

Giving a talk at Eli… on Academic Degrees and Sex Johan Aspegren on To Cheer You Up in Difficult T… To cheer you up in d… on Another sensation – Anni… Gil Kalai on To Cheer You Up in Difficult T… Gil Kalai on To Cheer You Up in Difficult T… Alexander Barvinok on To Cheer You Up in Difficult T… Kevin on To Cheer You Up in Difficult T… Gil Kalai on To Cheer You Up in Difficult T… Arseniy on To Cheer You Up in Difficult T… Alexander Barvinok on To Cheer You Up in Difficult T… uniform on To Cheer You Up in Difficult T… Arseniy on To Cheer You Up in Difficult T… -
### Recent Posts

- Giving a talk at Eli and Ricky’s geometry seminar. (October 19, 2021)
- To cheer you up in difficult times 32, Annika Heckel’s guest post: How does the Chromatic Number of a Random Graph Vary?
- To Cheer You Up in Difficult Times 31: Federico Ardila’s Four Axioms for Cultivating Diversity
- Dream a Little Dream: Quantum Computer Poetry for the Skeptics (Part I, mainly 2019)
- To Cheer you up in difficult times 30: Irit Dinur, Shai Evra, Ron Livne, Alex Lubotzky, and Shahar Mozes Constructed Locally Testable Codes with Constant Rate, Distance, and Locality
- To cheer you up in difficult times 29: Free will, predictability and quantum computers
- Alef’s corner: Mathematical research
- Let me tell you about three of my recent papers
- Mathematical news to cheer you up

### Top Posts & Pages

- Giving a talk at Eli and Ricky's geometry seminar. (October 19, 2021)
- Academic Degrees and Sex
- Answer: Lord Kelvin, The Age of the Earth, and the Age of the Sun
- The Argument Against Quantum Computers - A Very Short Introduction
- To Cheer You Up in Difficult Times 31: Federico Ardila's Four Axioms for Cultivating Diversity
- Richard Stanley: How the Proof of the Upper Bound Theorem (for spheres) was Found
- To cheer you up in difficult times 32, Annika Heckel's guest post: How does the Chromatic Number of a Random Graph Vary?
- Amazing: Karim Adiprasito proved the g-conjecture for spheres!
- TYI 30: Expected number of Dice throws

### RSS

### Categories

- Academics (14)
- Algebra (24)
- Analysis (14)
- Applied mathematics (4)
- Art (21)
- Blogging (12)
- Book review (4)
- Combinatorics (321)
- Computer Science and Optimization (134)
- Conferences (71)
- Controversies (1)
- Controversies and debates (22)
- Convex polytopes (61)
- Convexity (38)
- Economics (29)
- Education (1)
- Elections 2015 (7)
- Games (44)
- Geology (2)
- Geometry (69)
- Gina Says (8)
- Guest blogger (29)
- Happy birthday (3)
- ICM2018 (7)
- ICM2022 (1)
- Information theory (3)
- Law (6)
- Logic and set theory (2)
- Mathematical logic and set theory (2)
- Mathematics over the Internet (23)
- Mathematics to the rescue (13)
- Movies (7)
- Music (5)
- Number theory (23)
- Obituary (12)
- Open discussion (17)
- Open problems (100)
- People (7)
- personal (8)
- Philosophy (22)
- Physics (37)
- Poetry (10)
- Polymath10 (8)
- Polymath3 (10)
- Probability (84)
- Quantum (42)
- Rationality (31)
- Riddles (9)
- Sport (8)
- Statistics (8)
- Taxi-and-other-stories (22)
- Teaching (20)
- Test your intuition (62)
- Uncategorized (16)
- Updates (121)
- What is Mathematics (42)
- Women in science (15)

### Blogroll

- 4 Gravitons (Matt von Hippel)
- A cluttered mind (Deane Yang)
- Accidental Mathematician
- Affine mess (Alon Amit)
- Algorithmic Game Theory
- AMS open math notes
- Analysis of Boolean functions (Ryan O'Donnell)
- Andreas Caicedo’s teaching pages
- Anima on AI (Anima Anandkumar)
- Annoying Precision
- Anurag's math blog
- Area777
- Asaf Karagila
- Asymptotia
- Backreaction (Sabine Hossenfelder)
- Bits of DNA
- Bits of quantum
- Computational complexity
- Condenced concepts
- David Mumford
- Discrete analysis (the blog)
- Erdos problems on graphs
- Expositions and Riddles (Yuval Peres)
- Field arithmetician's blog
- Freedom Math Dance
- Geomblog
- Geometry and the Imagination
- Geometry at infinity
- Godel's lost letter and P=NP
- Gowers’s blog
- http://prelive.tricki.org/
- https://blogs.ams.org/beyondreviews/
- I am a Bandit (Sébastien Bubeck)
- Igor Pak's blog
- In theory
- John Baez
- Karim Adiprasito's blog
- Konard Swanepoel's Blog
- Kowalski’s Blog
- Krystal Guo
- Low dimensional topology
- Machine learning (theory)
- Math Overflow
- Math with bad drawings
- Mathbabe
- Mathemata
- Mathematical Enchantments
- Mathematical Gemstones
- Mathematical Musing
- Mathematics under the microscope
- Mathematics without apologies by Michael Harris
- Matt Baker Math Blog
- Michael Nielsen
- Misha Gromov home page
- MUSA The Gospel According to Math
- My Biased Coin
- My Qstate
- n-category cafe
- NeverEndingBooks
- njwildberger: tangential thoughts
- Noncommutative analysis
- Noncommutative geometry
- Not even wrong (Peter Woit)
- Numberphile
- Oded Goldreich's choices
- Open Problems Garden
- OXdE
- Persiflage – Galois representations and more
- Peter Cameron's Blog
- Peter Sarnak’s letters (and more)
- Piece of mind
- Quantum calculus
- Quantum Frontiers
- Quomodocumque
- Ratio Bound (Ferdinand Ihringer)
- Rigorous Trivialities
- Roots of unity
- Sarcastic resonance (Renan Gross)
- Sean Carroll
- Secret blogging seminar
- Short, fat matrices
- Shtetl Optimized
- Some plane truths (Adam Sheffer)
- Statistical Modeling
- Tanya Khovanova Math Blog
- TCS Math
- TCS stackexchange
- The Aperiodical
- The Geometry Junkyard
- The higher geometer
- The Intrepid mathematician (Anthony Bonato)
- The Polymath Blog
- The Quantum Pontiff
- Theorem of the day- complete listing
- Theory and theorems
- Theory of computing blog aggregator
- Thoughts by Emanuele Viola
- Tobias J. Osborne's research notes
- Tricky wiki
- What’s new (Terry Tao)
- Windows on Theory
- xkcd
- Yufei Zhao
- Zeros and Ones by Paata Ivanishvili
- דוד אסף על עניני מדינה ספרות ומדע – עונג שבת
- לא מדויק
- מדע גדול בקטנה
- נסיכת המדעים

### Archives

- October 2021 (2)
- September 2021 (3)
- August 2021 (4)
- July 2021 (1)
- June 2021 (2)
- May 2021 (4)
- April 2021 (3)
- March 2021 (5)
- February 2021 (8)
- January 2021 (8)
- December 2020 (6)
- November 2020 (6)
- October 2020 (4)
- September 2020 (3)
- August 2020 (5)
- July 2020 (2)
- June 2020 (1)
- May 2020 (3)
- April 2020 (4)
- March 2020 (6)
- February 2020 (5)
- January 2020 (7)
- December 2019 (7)
- November 2019 (1)
- October 2019 (5)
- September 2019 (7)
- August 2019 (7)
- July 2019 (7)
- June 2019 (1)
- May 2019 (3)
- April 2019 (4)
- March 2019 (9)
- February 2019 (7)
- January 2019 (2)
- December 2018 (4)
- November 2018 (1)
- October 2018 (4)
- September 2018 (3)
- August 2018 (2)
- July 2018 (2)
- June 2018 (8)
- May 2018 (2)
- April 2018 (8)
- March 2018 (3)
- February 2018 (6)
- January 2018 (8)
- December 2017 (5)
- November 2017 (6)
- October 2017 (11)
- September 2017 (6)
- August 2017 (3)
- June 2017 (1)
- May 2017 (3)
- April 2017 (4)
- March 2017 (4)
- February 2017 (5)
- January 2017 (6)
- December 2016 (1)
- November 2016 (1)
- October 2016 (2)
- September 2016 (1)
- August 2016 (1)
- July 2016 (2)
- May 2016 (5)
- April 2016 (5)
- March 2016 (1)
- January 2016 (2)
- December 2015 (1)
- November 2015 (2)
- October 2015 (3)
- September 2015 (1)
- August 2015 (4)
- May 2015 (2)
- April 2015 (4)
- March 2015 (7)
- February 2015 (4)
- January 2015 (2)
- December 2014 (8)
- November 2014 (2)
- October 2014 (2)
- September 2014 (1)
- August 2014 (4)
- July 2014 (3)
- June 2014 (3)
- May 2014 (3)
- March 2014 (2)
- February 2014 (2)
- January 2014 (3)
- November 2013 (2)
- October 2013 (3)
- September 2013 (9)
- August 2013 (3)
- July 2013 (3)
- June 2013 (3)
- May 2013 (10)
- April 2013 (8)
- March 2013 (9)
- February 2013 (2)
- January 2013 (4)
- December 2012 (5)
- November 2012 (3)
- October 2012 (1)
- August 2012 (2)
- July 2012 (2)
- June 2012 (5)
- May 2012 (2)
- April 2012 (5)
- March 2012 (4)
- February 2012 (2)
- January 2012 (2)
- December 2011 (4)
- November 2011 (3)
- October 2011 (2)
- September 2011 (2)
- August 2011 (3)
- July 2011 (3)
- June 2011 (4)
- April 2011 (1)
- February 2011 (3)
- January 2011 (6)
- November 2010 (8)
- October 2010 (9)
- September 2010 (1)
- August 2010 (1)
- July 2010 (1)
- June 2010 (4)
- May 2010 (3)
- April 2010 (1)
- March 2010 (3)
- February 2010 (10)
- January 2010 (9)
- December 2009 (7)
- November 2009 (5)
- October 2009 (1)
- September 2009 (3)
- August 2009 (9)
- July 2009 (12)
- June 2009 (11)
- May 2009 (12)
- April 2009 (12)
- March 2009 (10)
- February 2009 (10)
- January 2009 (11)
- December 2008 (13)
- November 2008 (12)
- October 2008 (5)
- September 2008 (8)
- August 2008 (6)
- July 2008 (8)
- June 2008 (13)
- May 2008 (11)
- April 2008 (1)

- Academics Algebra Analysis Art Combinatorics Computer Science and Optimization Conferences Controversies and debates Convexity Convex polytopes Economics Games Geometry Guest blogger Mathematics over the Internet Number theory Open discussion Open problems Philosophy Physics Probability Quantum Rationality Taxi-and-other-stories Teaching Test your intuition Uncategorized Updates What is Mathematics Women in science

- Alef's corner
- Alex Lubotzky
- Aram Harrow
- Avi Wigderson
- Boolean functions
- Borsuk's conjecture
- Branko Grunbaum
- Cap sets
- Combinatorics
- Conferences
- Convex polytopes
- Debates
- Discrepancy
- Dorit Aharonov
- Dor Minzer
- Ehud Friedgut
- Endre Szemeredi
- Eran Nevo
- Extremal combinatorics
- g-conjecture
- Games
- Game theory
- Greg Kuperberg
- Guy Kindler
- Günter Ziegler
- Helly type theorems
- Hirsch conjecture
- ICM2018
- Igor Pak
- Influence
- Itai Benjamini
- Jean Bourgain
- Jeff Kahn
- Karim Adiprasito
- Laci Lovasz
- Linear programming
- Lou Billera
- Mathoverflow
- Michal Linial
- Muli Safra
- Nati Linial
- Noam Lifshitz
- Noga Alon
- Noise
- Noise-sensitivity
- Oberwolfach
- Oded Schramm
- Paul Erdos
- Percolation
- Peter Frankl
- Peter Keevash
- polymath1
- Polymath3
- Polytopes
- Probability
- Quantum computation
- Quantum computers
- Quantum error-correction
- quantum supremacy
- Richard Stanley
- Robert Aumann
- Roth's theorem
- Scott Aaronson
- Sex
- sunflower conjecture
- Taxi-and-other-stories
- Terry Tao
- Test your intuition
- Tim Gowers
- Topological combinatorics
- Trees
- Turan's problem
- Tverberg's theorem
- Updates
- What is Mathematics

%d bloggers like this:

I would have thought bigger than two. I calculate three.

This seems so obvious — completely different from the usual “test your intuition” questions (like the previous one). What am I missing?

Who can say?! The answer seemed obvious to me for a few seconds, and then it stopped seeming obvious. Then a different answer became obvious, but only after some thinking 🙂

I had the same thing. I read the question, answered the poll with the first obvious answer (thinking first is cheating), thought for a few more seconds, and I am now sure that the second obvious answer is true.

Then I tried to make bad student’s mistakes and they were also options in the poll (one of them has place third in the current poll), so I think that there are at least three obvious answers.

As long as at each given moment there are no two contradictory obvious answers there is no global threat for mathematics as a whole.

Lior, I realize that it is important to have obvious questions from time to time so that readers’ intuition will not be tilted to the non-obvious.

Exactly!

A surprisingly nice question! What is the etiquette – can we spoil with comments down here, or should we wait a while…

Dear James, please comment on the new post coming soon.

Dear Lior and James, It is certainly easier than TYI 29. I shall reveal what people’s intuition was, and propose to wait for a detailed solution to the solution post.

Outcomes after 180 votes

Lovely results! I would love to know what kind of intuition leads someone to guess 1 here.

At least the truth comes in second.

By the way, the question was invented by Elchanan Mossel during a undergrad class. Students asked Elchanan to give more examples and he invented it on the spot (it was at Penn in 2014-15).

Actually while I gained your attention, I wonder if you regard the following exchange in a lecture funny or not:

Speaker: Aldous answered a conjecture of Mezard and Parisi and showed that in the limit the expectation for the assignment problem is .

Question from the audience: Did it confirmed or refuted the conjecture?

Another voice from the audience: Mezard and Parisi conjectured that the answer is not and Aldous refuted this conjecture!

Dear Gil,

A quibble. When I first read the question I was unsure if you meant a single FAIR die or a pair of FAIR dice were thrown. On rereading I decided you meant a single fair die.

Dear Joe, should I correct the English?

From the discussion I think everyone understood.

Yes, it is a single die rather than dice.

I thought dice is kosher for single as well.,,

Sorry, no dice 😉

What I saw was that: If you look up dice in the Oxford Dictionary, you will learn that dice is an acceptable singular and plural form of die. According to this source, dice was once the plural of die, “but in modern standard English dice is both the singular and the plural: ‘throw the dice’ could mean a reference to either one or more than one dice.”

Since I told Gil the story today, I will share the nicest explanation, that I heard from Paul Cuff.

Rephrase the question as follows: Toss a die until the first time T that the result is different from

2 and 4. What is the mean of T?

Finally, observe that T is independent of the value among 1,3,5,6 obtained in the T’th toss, so conditioning that value to be 6 does not affect the mean of T.

Pingback: Elchanan Mossel’s Amazing Dice Paradox (answers to TYI 30) | Combinatorics and more

Fantastic!

I can’t wait to unleash this on my students.

So what is the correct answer ?:)

I calculated 3….

That is a very elegant solution indeed. Here is a very different one – less compact, but with an ingredient that may be of general interest.

Consider a bet B on the number of throws: you get T dollars when the T-th throw is 6 after only 2s and 4s; as soon as a throw gives an odd number, the bet stops. Upfront you pay the mean payoff, call it x, but you get it back in the latter case. Then x is the answer to the puzzle.

It can be determined as follows. The payoff is

“1 if 6, x if 1,3,or 5, B’ if 2 or 4” (*)

Here B’ is the induced bet as of the second throw, “2 if 6, x if 1,3,or 5, B’’ if 2 or 4”, with B’’ the bet as of the third throw, etc.

Bet B’ has the same mean payoff as B+1/4: B’ pays 3/4 more when the last throw is 6 (probability 1/3), 1/4 less when it is odd (probability 3/4). So x must be the mean payoff of

“1 if 6, x if 1,3,or 5, x+1/4 if 2 or 4”,

and the solution follows.

Note: to replace B’ by x+1/4 is specific to the puzzle, but the idea behind using x in the payoff (*) relies on a very general idea: fixed point updating. It covers Bayesian updating as a special case, without reference to probabilities. The interested reader is referred to a post (August 18) on this topic at the Decision Theory forum at groups.google.com.

– Berend Roorda, University of Twente

I used to think “the die is cast” meant something like “the mold is set” — see Wikipedia’s “Die (manufacturing)” — rather than “the dice [sic] is thrown”. 🙂

This is a funny question. You have to define ‘expected’ and for a single die.

Then the number N for the number of throws will be in the range 1 < N < infinity (with very bad luck) and the probability curve peaking at pi!

Previous 2 and 4 throws don't count.

Pingback: Exploring Elchanan Moddel’s fantastic probability problem with kids | Mike's Math Page

Pingback: Exploring Elchanan Mossel’s fantastic probability problem with kids | Mike's Math Page

I got it wrong. Looking at the solution (where you ask about the first time you don’t get a 2 or a 4) I had a very nice ah ha/oohhhh experience. When I tried to go through it by figuring out the probability of getting an all even sequence I noticed something interesting. It’s 1/4, which means conditioned on getting only even numbers until a 6 the probability of getting a 6 on the first role is (1/6)/(1/4)=2/3. After the fact I can see why, if you get a 6 on the first roll then the game ends and you’re in the “all even” subworld, if you get a 2 or 4 you might (3/4 of the time) not be in the “all even” sequence world.

For intuition, does this remind anyone else of the Monty Hall problem? (Where realizing that conditioning really amounts to only caring about the average length of a die-generated sequence of 2’s and 4’s is sort of like realizing conditioning really amounts to only caring about whether the prize is behind the door that you originally picked.)

Really good problem! I makde a script to let everyone experiment and see why the correct answer is correct: http://www.airapport.com/2017/09/a-very-interesting-math-problem-with.html

It’s interesting that the answer doesn’t change, if you change the rule to stop on the first 5 or 6.

Pingback: A Probability Puzzle That You’ll Get Wrong | Math with Bad Drawings

Pingback: Counterintuitive Dice Probability: How many rolls expected to get a 6, given only even outcomes? | Dispatches from the Untrammeled Mind

Yuval’s elegant solution makes a creative leap to the problem of

throwing until a value other than two or four comes up, but I prefer a

simple approach that sticks to the original formulation of throwing a

six-sided fair die until the first six comes up.

Namely, the probability of the six-sided fair die throwing all evens

is unchanged if the first throw is two. That is, the probability of

throwing all evens is independent of the first throw being two. But

independence implies that conversely the probability of first throwing

two given all evens is the same as the probability of first throwing

two, namely 1/6; same for the probability of first throwing four given

all evens. It follows that the remaining possibility of first

throwing six given all evens has probability 1 – 1/6 – 1/6 = 2/3.

This reasoning applies equally well to the next throw after any

sequence of twos and fours given all evens, so throwing until a six

appears on the fair six-sided die, given that all throws are even,

is the same as throwing until a six appears on a three-sided die whose

probabilities of rolling two, four, and six are respectively 1/6, 1/6,

and 2/3. Of course the expected time for this unfair three-sided die

to roll a six is 1/(2/3) = 3/2.

Incorrect sketch ,if you hold the dice, focus on one side and count the spots, then look at the opposite side- count the spots and the total number of spots on those two sides should add up to 7

This is not a dice! The opposite faces must add to seven.

Pingback: Personal Learning and Development – Students into Education

Pingback: Talking conditional expectation with the boys thanks to examples from Alex Kontorovich and Gil Kalai – Mike's Math Page

Pingback: Understanding The Math Behind Elchanan Mossel’s Dice Paradox ~ Mathematics ~ mathubs.com