Dice
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.
If you roll two dice, what is the probability the sum of the numbers you get is odd?
Answer
1/2
The first die you roll  whether it's even or odd doesn't matter. The second die you roll has to be the opposite.
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Comments
vlerma
Apr 04, 2006
 How many times did you toss the dice? I think you would have to specify how many times one could roll the dice to come up with your answer. It is still just the luck of the roll. Cute idea tho. 
zonarita
Apr 04, 2006
 Smarty this didn't work when I started rolling die. I might not understand your explanation. I like anything that is die related.
(I rolled double sixes, followed by two ones; followed by a four and five and a pair of twos. I rolled for a while and was pretty shocked at the pairs I kept rolling. Never got a pair of fours  oh well.) 
brainjuice
Apr 05, 2006
 i think the answer is 1/4.. i get it wrong.. wew.. i really like dice things .. haha.. 
cms271828
Apr 05, 2006
 Nice solution, its perfectly correct.
Basically the parity of the second dice must be different to the parity of the first dice, so it has a 50,50 chance.
Hence probabililty is half.
For those who don't believe it, just draw a possibility space.
...1.2.3.4.5.6

12.3.4.5.6.7
23.4.5.6.7.8
34.5.6.7.8.9
45.6.7.8.9.10
56.7.8.9.10.11
67.8.9.10.11.12
Half the numbers are odd, and all 36 cases are equally likely, so probability is half 
thecelticgod
Apr 06, 2006
 Way to go CMS. Perfect explanation of probability and proof of the answer. Remember, probability is constant, short term runs in either direction are possible. 
Herman
Apr 06, 2006
 No, the probability is 44.44%. If you actually count out all the combinations of two dice, and count the number of ways to get odds, it comes out differently. 
cms271828
Apr 06, 2006
 All the combinations are listed in the possibility space above, so its 50%.
Or use the other proof that the parity of the second dice must be the opposite parity of the first.
Either way, its 50%. 
cms271828
Apr 06, 2006
 Herman, it appears you miscounted, and counted 16 out of 36 had an odd sum.
It should be 18 out of 36. 
calmsavior
Apr 07, 2006
 i actually made CMS' chart to solve this as well. 
Lui312
Apr 08, 2006
 Can't be right!How the heck is that possible....!!!!!!!!!!!!!!!!!!!!!!!! 
brainjuice
Apr 09, 2006
 lui, it is correct. look at cms' chart.. the answer is 18/36 = 1/2 
charles445
Apr 20, 2006
 This one has a mathematical spring to it, so i like it!
I made a chart like cvs's anyways to figure it out. 
Jimbo
Apr 26, 2006
 Another commonsense way to think about it is that both dice are equally likely to come up either odd or even so its just like flipping 2 coins marked odd and even. Now ODD + ODD = EVEN and EVEN + EVEN = EVEN and ODD + EVEN = ODD and EVEN + ODD = ODD so 2 out of the four ways the coins could fall make EVEN and 2 of the four ways make ODD so 50% is the answer. In other words since the numbers on each die are expected to turn up odd or even on a 50/50 basis and the sums of LIKE/UNLIKE turn out odd or even on a 50/50 basis then we don't expect any bias in the resulting sum! 
eamon
Oct 16, 2006
 I had to work this one out the slow way with a pencil listing all possible combinations to get the answer. It surprised me a bit as I never have done much with dice probabilities before. The statement "The second die you roll has to be the opposite." is hardly correct if you think about it, but the final answer of 1/2 agreed with mine.
Nice one! 
ztodd
Oct 21, 2006
 I believe what was meant was that if you want the sum to be odd then the second die you roll has to be the opposite parity of the first. 
billb
Nov 17, 2006
 It is Not a simple 1/2. Rolling 2 dice will give you a total of between 2 and 12, with possibility for 6 even totals and 5 odd totals so the simple odds are 6 to 5 in favor of even 
billb
Dec 02, 2006
 Part 2. You asked about the SUM in your question, Not the odds of Matching, which are 1/2.
Probability on the SUM being ODD are 6/5 against there are 6 chances for an EVEN sum and 5 chances of an ODD sum on any roll of 2 Dice. 
Whitley
Dec 17, 2006
 BillB One would THINK that... But it's not true. There are Six even sums, and five odd sums, but there's more ways to GET the odd sums than there are the evens. For instance, the only way you can get 2 is 1+1, and the only way you can get 12 is 6+6, but you can get seven three different ways. (6+1, 5+2, 3+4)
So it does balance out to be a fifty percent chance. Here's my chart all the odd possibilities have a star.
1+1=2
1+2=3*
1+3=4
1+4=5*
1+5=6
1+6=7*
2+1=3*
2+2=4
2+3=5*
2+4=6
2+5=7*
2+6=8
3+1=4
3+2=5*
3+3=6
3+4=7*
3+5=8
3+6=9*
4+1=5*
4+2=6
4+3=7*
4+4=8
4+5=9*
4+6=10
5+1=6
5+2=7*
5+3=8
5+4=9*
5+5=10
5+6=11*
6+1=7*
6+2=8
6+3=9*
6+4=10
6+5=11*
6+6=12
Probability: 18/36 
Eshootzi_scrs
Apr 25, 2007
 Ok I get the idea but somebody tell me why this isnt correct.
There are ACTUALLY only 21 combinations of the two dice.
If you roll two dice it would be assumed you rolled them together, otherwise you could roll one twice.
So the combination 2 1 is the same as the combination 1 2, both have an odd sum but equal one combinbation not two.
Following then the combinations factor to 6 + 5 + 4 + 3 + 2 + 1 = 21.
Of the combinations 9 have odd sums and 12 have even sums.
Therefore you are 9 of 21 to be even or 9 : 12.
How many two card combinations are there in a standard 52 card deck for Texas Hold'em? 
tsimkin
Apr 27, 2007
 I assume you are kidding when you say there are 21 possible combinations and not 36. Image one die is red and the other is white. A red 2 and a white 1 is a distinctly different roll from a red 1 and a white 2. To use your betting mechanism from another comment you posted, I would be more than happy to play you 100 times, where you pay me $1.10 if it is even, and I will pay you $1.00 if it is odd. (And to answer your HE question, there are 1326 distinct starting hands.) 
late664
Jun 11, 2007
 Okay, after reading the answer and comments, I understand the nature of the question.
However, upon first reading it... I was under the impression that '"if you roll 2 dice", it meant that you rolled both at the same time.
If that was the case... then the possible sum of both dice would be 
2,3,4,5,6,7,8,9,10,11,and 12
5 possible odds, and 6 possible evens. 
tsimkin
Jun 13, 2007
 late664  those are indeed the possible sums, but they are not all equally possible. A 7 comes up 6 times, while a 12 only comes up once. You have to take this into account, as well. Otherwise, you could look at your results from buying a lottery ticket as either being down $1 or up $1,000,000, and conclude that you are therefore 50% to win a million! 
opqpop
Jan 27, 2010
 This is way too easy. I read the hint and I don't understand how one can even come up with 1/3 
opqpop
Sep 16, 2010
 Wow I must have gone dumber. I did 2 * 1/4 this time via casework odd + even and even + odd instead of fixing the first roll.... 
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