Dween
Joined: Jan 24, 2010
• Posts: 339
October 21st, 2013 at 10:01:16 AM permalink
Along the vein of the Two Coin Puzzle, I wanted to put this in another thread as to not clog up the other.

You have two 6-sided dice in a cup. You shake the dice, and slam the cup down onto the table, hiding the result. Your partner peeks under the cup, and tells you, truthfully, "At least one of the dice is a 2."

What is the probability that both dice are showing a 2?

Based on the discussion in the Two Coin thread, I'm interested to see how people solve this one.

Pleased to meet you, name's Francis Pumphandle, but everyone calls me Pip. Cheese balls are one of my all-time favorite foods. I always seem to meet the most interesting people when I'm around them, too. In fact, cheese balls bring to mind the time I met Bob Barker. Yes, indeed -- Bob Barker, star of the most popular morning game show. He's an emcee, a host and a celebrity all rolled into one. Anyway, eight months ago, it was Tuesday the 17th, I believe, or it might have been the 18th -- no, no, it was definitely the 17th, because it was precisely one week after my Aunt Lucretia's birthday, which is the 10th. Aunt Lucretia's quite a woman -- loves to cook. She prepares a fabulous "war-shoo-off" -- that's a Chinese duck dish. I love Chinese food. I once went to a party where they served Chinese food and cheese balls. Now that was a Catch-22 situation. Catch-22 was a movie, you know. It was long -- VERY long. They say the book was better, but it was a novel, and I never finish reading those things. Of course, a lot of people don't read much nowadays; they watch television. I caught a program on PBS last night -- a very good show on chimpanzees in the media. They had a clip of Jay Fred Mug and a chimp on the Today Show, but it was Fred's chimpanzee's girlfriend that had me stumped. I couldn't remember her name, so I looked it up. Her name was Fibi B. Bibi. Anyway, as I was saying, eight months ago, Tuesday the 17th, I went downtown on a nice relaxing stroll. I love to relax. In fact, relaxing is a hobby of mine. Some people play golf, others like tennis, horseshoes, bridge, canasta, and other such fancy hobbies. Now another hobby enjoyed by many is knitting. My grandmother was a great knitter -- knitted this sweater I'm wearing. It's red, which is not my favorite color. I prefer mauve or a mustard yellow. Now, don't get me wrong, red is o.k. for ties and suspenders, but with sweaters I prefer more neutral colors. But when I'm relaxing, I don't care WHAT I wear -- long pants, bermuda shorts, t-shirts, or formal attire, you name it -- anything goes. Now, on the 17th, during my relaxing stroll, I recall wearing my herringbone jacket, my Laughlin, Nevada souvenir tie, and my charcoal grey slacks -- or was it the navy slacks? Oh, I suppose it doesn't really matter, does it? What matters is comfort. You know, I once stayed at a Comfort Inn -- warm, cozy, comfortable. I love comfort. It goes along with that pastime of mine -- relaxing. Now, for me, there's nothing more relaxing than a nice leisurely stroll like the one I took eight months ago on the 17th. It was a bright, sunny day, which of course is the optimum condition for relaxed strolling. And as I walked along, I found myself humming a haunting melody. I kept humming and humming and humming and humming. I couldn't get the tune out of my head. I racked my brain to come up with the title, but to no avail. You see, I'm not terribly musical -- and yet, I'd always wanted to play a musical instrument and be like my musical hero, Leo Sayer. But who can compete with Leo? I think I was just scared I'd fail. Well, I decided right then and there to go buy a musical instrument. So on the particular Tuesday the 17th to which I was referring, I went down to the Sixth Street Musical Emporium to buy a new tambourine, a terribly soothing instrument contrary to popular opinion. And as I was strolling along, I detected a wonderful scent in the morning air. "What could it be?" I asked myself. So I went toward that marvelous scent, distracted by its aroma from my musical mission. The odor was a mix of orchid flowers and bologna, which, of course, is one of the world's most underappreciated luncheon meats -- that and pimento loaf. I love a good pimento-loaf-and-mayo sandwich -- the more pimentos, the better. Why just the mention of pimentos makes my taste buds stand up and say "Howdy". Now there's an interesting word -- "howdy". Is it from "How are you?", or maybe "How ya doing?" "Howdy"'s one of those strange words that really HAS no origin. I like saying, "How do" more than "Howdy" -- more formal, I think -- not too flowery. But the flowery aroma of that particular Tuesday morning carried me on my fragrant quest. Now, the smell was actually less bologna and more orchid, the beautiful flower found on the island state of Hawaii. Of course, I wasn't in Hawaii, so I needed to search out the location of the nearest orchid. So, I visited every flower shop in town. Well, to make a long story short, not a SINGLE flower shop in town had ANY orchids in stock, which seemed mighty curious to me. Now, as we all know, curiosity killed the cat, but since I'm not feline, I wasn't too worried. Felines are funny creatures, don't you think? I had a cat once. It used its claws to tear my living room couch to shreds. It was a comfy couch, too -- had a sleepaway bed in it with a foam rubber mattress. Now, I bought the couch AND the mattress at Levine's department store on Third Avenue the very same afternoon of that relaxing stroll aforementioned. I also bought myself a lovely tambourine on that same shopping expedition. Anyway, I didn't want to pay extra for the delivery of the couch, so I decided to carry the couch home myself. It was quite cumbersome, and getting it through the store's revolving doors was a bit of a challenge. And just as I emerged onto the street, by accident I bumped into a well-dressed man with an orchid in his lapel. It was Bob Barker, and he was eating and bologna-and-cheese-ball sandwich. Well, it's been nice chatting with you. Bye!
-Dween!
Wizard
Joined: Oct 14, 2009
• Posts: 24362
October 21st, 2013 at 10:22:05 AM permalink
pr(two sixes)/pr(one or more six) = (1/36)/(11/36) = 1/11
It's not whether you win or lose; it's whether or not you had a good bet.
MathExtremist
Joined: Aug 31, 2010
• Posts: 6526
Thanks for this post from:
October 21st, 2013 at 10:57:07 AM permalink
The Wizard is correct under the assumption that both dice are standard six-sided dice with 1..6 on them, instead of something like:
Last edited by: unnamed administrator on Oct 4, 2016
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice." -- Girolamo Cardano, 1563
MangoJ
Joined: Mar 12, 2011
• Posts: 905
October 21st, 2013 at 1:39:00 PM permalink

... 36 combinations of two 6-sided dices, only 11 of them show at least a 2, from which one is 2-2. Hence the probability would be 1/11.

..this would only hold if the player would say nothing when no 2 were spotted.

If (before peeking) your friend is determined to say "at least one of the dice is a X" where X is one of the numbers he may spots, the probability for a pair would be different.

If the friend sees any pair (1/6), probability of "X-X" where X is the named number, is one.
If the friend sees a non-pair (5/6), probability of "X-X" is zero whatever he choses for X.

Hence, like the Monty-Hall-Problem, if your friend must always say "At least one of the dice is a X" (while not lying), probability of a pair is 1/6.
indignant99
Joined: Feb 23, 2015
• Posts: 250
April 13th, 2015 at 1:28:17 PM permalink
Short answer: 1/6 = 16.667% probability.
(God damn it, Michael, we've ruled OUT a bunch of your denominators.)
I love showing the Wizard wrong!

ProbDie-1Die-2Note
1/3621
2/362 2
1/3623
1/3624
1/3625
1/3626
ProbDie-1Die-2Note
1/3612
022 Already accounted for above (pair of deuces)
1/3632
1/3642
1/3652
1/3662

One-Third of all possible rolls have a deuce (or two of them) in them.
Two-thirds of all possible rolls lack any deuce.

Out of the 12 rolls that contain a deuce (or two deuces), Two have a pair of deuces.
Two out of twelve = 1/6 = 16.667% Period, the end.
Yeah, I made a mistake once. I thought I was wrong, when I actually wasn't. -Indignant
ThatDonGuy
Joined: Jun 22, 2011
• Posts: 5417
April 13th, 2015 at 2:12:10 PM permalink
Quote: indignant99

Short answer: 1/6 = 16.667% probability.
(God damn it, Michael, we've ruled OUT a bunch of your denominators.)
I love showing the Wizard wrong!

And perhaps some day, you will...but it's not today.

The probability of throwing a pair of 2s is not 1/18.

Think about it. Let's change the problem from 2s to 6s. According to you, the probability of throwing a pair of 6s is 1/18. Pretty much every craps table in Vegas offers 29-1 or better on this. Talk about an advantage play!

In your table, change "die 1" to "red die" and "die 2" to blue die. 6 out of 36 rolls will have the red die be 2, and of those 6, only one will have the blue die be 2 as well.
surrender88s
Joined: Jun 23, 2013
• Posts: 291
April 13th, 2015 at 2:22:07 PM permalink
In this problem, the probability is not the same as rolling two dice since you have new data. I think 1/6 may be correct, since you have truthful knowledge that at least one di is already a 2.
"Rule No.1: Never lose money. Rule No.2: Never forget rule No.1." -Warren Buffett on risk/return
indignant99
Joined: Feb 23, 2015
• Posts: 250
April 13th, 2015 at 2:47:28 PM permalink
How do you repress the occurrence of "2" on the unknown die, to
1=twice
2=once
3=twice
4=twice
5=twice
6=twice
(one out of 11)? Absurd.
Yeah, I made a mistake once. I thought I was wrong, when I actually wasn't. -Indignant
ThatDonGuy
Joined: Jun 22, 2011
• Posts: 5417
April 13th, 2015 at 2:50:00 PM permalink
Quote: surrender88s

In this problem, the probability is not the same as rolling two dice since you have new data. I think 1/6 may be correct, since you have truthful knowledge that at least one di is already a 2.

What do you mean, "you have new data"?

Either you have "at least one 2", or you have zero 2s, so the probability of at least one 2 = 1 - the probability of zero 2s = 1 - 25/36 = 11/36. In other words, 11 out of the 36 rolls have one 2. How many of these 11 have two 2s?
ThatDonGuy
Joined: Jun 22, 2011
• Posts: 5417
April 13th, 2015 at 2:59:55 PM permalink
Quote: indignant99

How do you repress the occurrence of "2" on the unknown die, to
1=twice
2=once
3=twice
4=twice
5=twice
6=twice
(one out of 11)? Absurd.

Either die can be "the unknown die".

Let one of the dice be red, and the other blue.
The columns in this table refer to the blue die, and the rows to the red die.
Red Die123456
1010000
2121111
3010000
4010000
5010000
6010000

As you can see, 11 of the 36 rolls have at least one 2; of these, only one has two 2s.

Probability = 1 / 11.