A roll of the dice

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A roll of the dice

Postby Gord » Fri Nov 06, 2009 9:13 pm

Here's one for your statistical analysis:

I'm sitting at a table. You walk in and hand me a pair of dice. Each die has six sides, and you are certain that they are fair -- in other words, they aren't loaded at all, and they would be acceptable to anyone who wanted to roll fairly.

You sit down across the table from me. Then I roll both dice behind a screen where you can't see them.

"Did you roll at least one 1?" you ask me.

I look at the dice. "Yes," I reply. "At least one of these two dice shows a 1 on top."

Now here's the question: What are the odds that I rolled two 1s? :mrgreen:
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Re: A roll of the dice

Postby Paul » Fri Nov 06, 2009 10:08 pm

Gord wrote:Here's one for your statistical analysis:

I'm sitting at a table. You walk in and hand me a pair of dice. Each die has six sides, and you are certain that they are fair -- in other words, they aren't loaded at all, and they would be acceptable to anyone who wanted to roll fairly.

You sit down across the table from me. Then I roll both dice behind a screen where you can't see them.

"Did you roll at least one 1?" you ask me.

I look at the dice. "Yes," I reply. "At least one of these two dice shows a 1 on top."

Now here's the question: What are the odds that I rolled two 1s? :mrgreen:

You have no spoiler. :frown: My horrible guess is 1-20 and I don't know Why.
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Re: A roll of the dice

Postby Gord » Fri Nov 06, 2009 10:21 pm

Paul wrote:You have no spoiler. :frown: My horrible guess is 1-20 and I don't know Why.

I have no spoiler 'cause I want to find The Person Who Can Get This Right.

...but don't worry it's not you. :mrgreen: Your "horrible guess" is wrong.
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Re: A roll of the dice

Postby Paul » Fri Nov 06, 2009 11:43 pm

Gord wrote:
Paul wrote:You have no spoiler. :frown: My horrible guess is 1-20 and I don't know Why.

I have no spoiler 'cause I want to find The Person Who Can Get This Right.

...but don't worry it's not you. :mrgreen: Your "horrible guess" is wrong.


1 in 30 Did I do good..... :roll:
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Re: A roll of the dice

Postby Hotair101 » Sat Nov 07, 2009 9:32 am

6/1.
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Re: A roll of the dice

Postby Gord » Sat Nov 07, 2009 10:54 am

Paul wrote:1 in 30 Did I do good..... :roll:

Yes, you did good...but you still got it wrong.

Hotair101 wrote:6/1.

Nope, it's not one in six either.
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Re: A roll of the dice

Postby vanderpoel » Sat Nov 07, 2009 11:02 am

1:36
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Re: A roll of the dice

Postby Gord » Sat Nov 07, 2009 11:24 am

vanderpoel wrote:1:36

Nope! :mrgreen:
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Re: A roll of the dice

Postby Paul » Sat Nov 07, 2009 12:20 pm

Gord wrote:
Paul wrote:1 in 30 Did I do good..... :roll:

Yes, you did good...but you still got it wrong.

Hotair101 wrote:6/1.

Nope, it's not one in six either.


No, I cheated and still got it wrong,what's up with that? :oops:
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Re: A roll of the dice

Postby Bishadi » Sat Nov 07, 2009 1:30 pm

100%

if both dice 'rolled'

both 1's rolled
.


IF existence operates one ONE way, is the math the 'name' to know?

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Re: A roll of the dice

Postby Gord » Sat Nov 07, 2009 7:30 pm

Paul wrote:No, I cheated and still got it wrong,what's up with that? :oops:
http://www.oldhatcreative.com/blog/wp-c ... -table.jpg

I don't think craps odds will work here. In craps, you don't usually know that one of the dice will be a 1 before you bet.

Bishadi wrote:100%

if both dice 'rolled'

both 1's rolled

No, "rolled two 1s" means the 1 is on the top of each respective die when it comes to rest.

:snake-eyes:
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Re: A roll of the dice

Postby Bishadi » Sat Nov 07, 2009 10:51 pm

Gord wrote:
Bishadi wrote:100%

if both dice 'rolled'

both 1's rolled

No, "rolled two 1s" means the 1 is on the top of each respective die when it comes to rest.

:snake-eyes:


lots of people change, as they get educated (you evolving, as the environment is much different now)

been awhile since you came across a mind :lol:
.


IF existence operates one ONE way, is the math the 'name' to know?

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Re: A roll of the dice

Postby Gord » Sun Nov 08, 2009 1:30 am

Bishadi wrote:lots of people change, as they get educated (you evolving, as the environment is much different now)

I'm not evolvin', I'm dyin'. Slowwwwly, but surrrrely.

been awhile since you came across a mind :lol:

That's gross. :P
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Re: A roll of the dice

Postby fromthehills » Sun Nov 08, 2009 2:34 pm

1 in 12

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Re: A roll of the dice

Postby Bishadi » Sun Nov 08, 2009 2:53 pm

Gord wrote:
Bishadi wrote:lots of people change, as they get educated (you evolving, as the environment is much different now)

I'm not evolvin', I'm dyin'. Slowwwwly, but surrrrely.


nothing random about that; it's your choice

perhaps life is nothing but "taking" to you and soon will not be able to take, by choice

but to live, give (ever notice how Darwin WON?)

Gord wrote:
been awhile since you came across a mind :lol:

That's gross. :P


anyon can see, that thinking and observing others over yourself, is not normally your mental occupation

kind of like the fact that many will not pursue their red wings
.


IF existence operates one ONE way, is the math the 'name' to know?

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Re: A roll of the dice

Postby Gord » Sun Nov 08, 2009 3:15 pm

fromthehills wrote:1 in 12

Nope!
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Re: A roll of the dice

Postby Gord » Sun Nov 08, 2009 3:18 pm

Bishadi wrote:
Gord wrote:
Bishadi wrote:lots of people change, as they get educated (you evolving, as the environment is much different now)

I'm not evolvin', I'm dyin'. Slowwwwly, but surrrrely.


nothing random about that; it's your choice

Death may not be random, but it may also not be a choice either.

perhaps life is nothing but "taking" to you and soon will not be able to take, by choice

Also, perhaps not.

but to live, give (ever notice how Darwin WON?)

Darwin died.

Gord wrote:
been awhile since you came across a mind :lol:

That's gross. :P


anyon can see, that thinking and observing others over yourself, is not normally your mental occupation

I hope you'll understand if I remind you that I believe your judgment is flawed.

kind of like the fact that many will not pursue their red wings

I don't even LIKE hockey.
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Re: A roll of the dice

Postby fromthehills » Sun Nov 08, 2009 3:26 pm

1 in 648

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Re: A roll of the dice

Postby Gord » Sun Nov 08, 2009 3:28 pm

fromthehills wrote:1 in 648

You're just guessing randomly now! :P

I'll just tell you the answer: It's 1 in 11.
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Re: A roll of the dice

Postby fromthehills » Sun Nov 08, 2009 3:32 pm

That wasn't random. I multiplied 36 by itself and divided by 2.

I get it. My 1 in 12 was flawed by the fact that one was given.

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Re: A roll of the dice

Postby Paul » Sun Nov 08, 2009 3:32 pm

At least one of these two dice shows a 1 on top."
If you can see the 1 on top, then the odds should be

:beg:
1 in 5------- final answer
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Re: A roll of the dice

Postby Gord » Sun Nov 08, 2009 3:49 pm

Paul wrote:At least one of these two dice shows a 1 on top."
If you can see the 1 on top, then the odds should be

:beg:
1 in 5------- final answer

No, it's 1 in 11.

Here's how to understand it (in the simplest way I can come up with):

  • When I rolled the two dice, there were 36 possible combination. That's because each die has six sides; and when you combine odds in this particular way, you multiply the individual numbers of possible outcomes together to find the combined number of possible outcomes.
  • When I told you at least one of them showed a 1, you have to discard all the combinations that don't contain at least one 1. It turns out, there are 25 of them -- 5 on one die by 5 on the other die, 5 x 5 = 25.
  • 36 - 25 = 11. There are 11 possible combinations remaining.
  • Only one of those combinations contains two 1s: Snake-eyes. When I first rolled, there was a 1 in 36 chance to roll two 1s, and then we discounted 25 of those chances from having happened.
  • That leaves 1 chance out of 11 remaining possible outcomes.

Ta-dah!
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Re: A roll of the dice

Postby Paul » Sun Nov 08, 2009 6:18 pm

Gord wrote:
Paul wrote:At least one of these two dice shows a 1 on top."
If you can see the 1 on top, then the odds should be

:beg:
1 in 5------- final answer

No, it's 1 in 11.

Here's how to understand it (in the simplest way I can come up with):

  • When I rolled the two dice, there were 36 possible combination. That's because each die has six sides; and when you combine odds in this particular way, you multiply the individual numbers of possible outcomes together to find the combined number of possible outcomes.
  • When I told you at least one of them showed a 1, you have to discard all the combinations that don't contain at least one 1. It turns out, there are 25 of them -- 5 on one die by 5 on the other die, 5 x 5 = 25.
  • 36 - 25 = 11. There are 11 possible combinations remaining.
  • Only one of those combinations contains two 1s: Snake-eyes. When I first rolled, there was a 1 in 36 chance to roll two 1s, and then we discounted 25 of those chances from having happened.
  • That leaves 1 chance out of 11 remaining possible outcomes.

Ta-dah!

Once you have said you can see the one die already has a one, you are only now concerned with the odds that the other die contains a one as well. At that point you are only dealing with the odds of the remaining die. For that die its 1 out of 5. And you are mixing the words chance with odds. Chance is a completley different set of mathematical equations. Maybee in Canada you can do it that way ....but its not the correct way. :mrgreen:
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Re: A roll of the dice

Postby OlegTheBatty » Sun Nov 08, 2009 9:23 pm

Paul wrote:
Gord wrote:
Paul wrote:At least one of these two dice shows a 1 on top."
If you can see the 1 on top, then the odds should be

:beg:
1 in 5------- final answer

No, it's 1 in 11.

Here's how to understand it (in the simplest way I can come up with):

  • When I rolled the two dice, there were 36 possible combination. That's because each die has six sides; and when you combine odds in this particular way, you multiply the individual numbers of possible outcomes together to find the combined number of possible outcomes.
  • When I told you at least one of them showed a 1, you have to discard all the combinations that don't contain at least one 1. It turns out, there are 25 of them -- 5 on one die by 5 on the other die, 5 x 5 = 25.
  • 36 - 25 = 11. There are 11 possible combinations remaining.
  • Only one of those combinations contains two 1s: Snake-eyes. When I first rolled, there was a 1 in 36 chance to roll two 1s, and then we discounted 25 of those chances from having happened.
  • That leaves 1 chance out of 11 remaining possible outcomes.

Ta-dah!

Once you have said you can see the one die already has a one, you are only now concerned with the odds that the other die contains a one as well. At that point you are only dealing with the odds of the remaining die. For that die its 1 out of 5. And you are mixing the words chance with odds. Chance is a completley different set of mathematical equations. Maybee in Canada you can do it that way ....but its not the correct way. :mrgreen:


Which die is a known 1, Paul, die a or die b? Its an unknown, therefore all combinations where die a is a 1 and all combinations where die b is a 1 must be considered.
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Re: A roll of the dice

Postby Gord » Mon Nov 09, 2009 1:52 am

Paul wrote:Once you have said you can see the one die already has a one, you are only now concerned with the odds that the other die contains a one as well. At that point you are only dealing with the odds of the remaining die.

What Oleg said. The two dice are individual entities, and either one would be the die I'm looking at. For instance, if one die is red and the other blue, here are the possibilities that remain when I say at least one of them rolled a 1:

1 and 1
1 and 2
2 and 1
1 and 3
3 and 1
1 and 4
4 and 1
1 and 5
5 and 1
1 and 6
6 and 1

For that die its 1 out of 5.

I hope you meant 1 out of 6. :mrgreen:
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Re: A roll of the dice

Postby Martin Brock » Mon Nov 09, 2009 4:36 am

Paul wrote:Once you have said you can see the one die already has a one, you are only now concerned with the odds that the other die contains a one as well. At that point you are only dealing with the odds of the remaining die.

Which of the two dice is the remaining die? You can't answer this question, and that's the problem with your analysis. The die seen to display 1 could be either die, and which die it is specifically is unknown to you. You only know that at least one of two dice display 1.

You're thinking of a different trial in which I roll one die, tell you that it landed on 1 and then ask you the odds that a second die also lands on 1, but that's not the same trial. In Gord's trial, the second die I roll could be the die he identifies as displaying 1.
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Re: A roll of the dice

Postby vanderpoel » Mon Nov 09, 2009 7:27 am

Gord wrote:When I first rolled, there was a 1 in 36 chance to roll two 1s,

You asked: "What are the odds that I rolled two 1s?
My answer: 1 in 36
Ta dah!
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Re: A roll of the dice

Postby Hotair101 » Mon Nov 09, 2009 8:51 pm

Yeah, and if one die has already stopped on a 1 then the odds of the last die stopping on a one are 6/1.
Ta dahhhh!!!!! lol.
P.s. I think you brain teased yourself lol.
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Re: A roll of the dice

Postby Gord » Tue Nov 10, 2009 4:39 am

vanderpoel wrote:
Gord wrote:When I first rolled, there was a 1 in 36 chance to roll two 1s,

You asked: "What are the odds that I rolled two 1s?
My answer: 1 in 36
Ta dah!

Nope. After I tell you that I rolled at least one 1, the chances of having rolled two 1s changes. It's now 1 in 11.

Hotair101 wrote:Yeah, and if one die has already stopped on a 1 then the odds of the last die stopping on a one are 6/1.
Ta dahhhh!!!!! lol.
P.s. I think you brain teased yourself lol.

Yes, and if the other die has already stopped on a 1, then the odds of the first die stopping on a one are also 1/6 (I assume that's what you meant by "6/1"). You then have to combine the two odds, with the consideration that at least one of the did roll a 1. The resulting chance comes out to 1 in 11.
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Re: A roll of the dice

Postby vanderpoel » Tue Nov 10, 2009 8:49 am

Gord wrote:
vanderpoel wrote:
Gord wrote:When I first rolled, there was a 1 in 36 chance to roll two 1s,

You asked: "What are the odds that I rolled two 1s?
My answer: 1 in 36
Ta dah!

Gord wrote:Nope. After I tell you that I rolled at least one 1, the chances of having rolled two 1s changes. It's now 1 in 11.

No, your question refers to a past tense occurrence. It refers to the odds when you first rolled. The fact that you rolled a 1 does not alter the odds. Besides, for all we know there might actually be another 1 on the other die. As a matter of fact the indication that you rolled "at least" one allows for that possibility. But what are the odds that you might disagree with this finding? :roll:
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Re: A roll of the dice

Postby Hotair101 » Tue Nov 10, 2009 5:13 pm

vanderpoel wrote:
Gord wrote:
vanderpoel wrote:
Gord wrote:When I first rolled, there was a 1 in 36 chance to roll two 1s,

You asked: "What are the odds that I rolled two 1s?
My answer: 1 in 36
Ta dah!

Gord wrote:Nope. After I tell you that I rolled at least one 1, the chances of having rolled two 1s changes. It's now 1 in 11.

No, your question refers to a past tense occurrence. It refers to the odds when you first rolled. The fact that you rolled a 1 does not alter the odds. Besides, for all we know there might actually be another 1 on the other die. As a matter of fact the indication that you rolled "at least" one allows for that possibility. But what are the odds that you might disagree with this finding? :roll:

100 per cent, lol, because it is wrong. You are rolling two dice, Van, and you have just tallied the number of possible outcomes. Where you are going wrong is that in the 36 possible outcomes some numbers have a better chance of being rolled than others.
For example the numbers 2 and 12 can only be rolled by either rolling two 1's or two sixes. This is the only way to roll these numbers. The number 7 can be rolled six ways. 1/6, 2/5, 3/4, 6/1, 5/2, 4/3 giving it a better probability of being rolled.

I think your question could have been phrased better, Gord. This is your question.
Now here's the question: What are the odds that I rolled two 1s?

The odds on you rolling snakes eyes are one in six.

But I don't think this is the question you are asking?
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Re: A roll of the dice

Postby Gord » Tue Nov 10, 2009 9:23 pm

vanderpoel wrote:
Gord wrote:
vanderpoel wrote:
Gord wrote:When I first rolled, there was a 1 in 36 chance to roll two 1s,

You asked: "What are the odds that I rolled two 1s?
My answer: 1 in 36
Ta dah!

Gord wrote:Nope. After I tell you that I rolled at least one 1, the chances of having rolled two 1s changes. It's now 1 in 11.

No, your question refers to a past tense occurrence. It refers to the odds when you first rolled. The fact that you rolled a 1 does not alter the odds.

After I tell you that I rolled at least one 1, the possibilities of what I rolled are altered. For example, you now know I didn't roll a pair of 2s.

Besides, for all we know there might actually be another 1 on the other die. As a matter of fact the indication that you rolled "at least" one allows for that possibility.

That's why it's a brain teaser!

But what are the odds that you might disagree with this finding? :roll:

I could agree with you if you really want that, but then we'd both be wrong. :pardon:
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Re: A roll of the dice

Postby Gord » Tue Nov 10, 2009 9:35 pm

Hotair101 wrote:Where you are going wrong is that in the 36 possible outcomes some numbers have a better chance of being rolled than others.

Actually, the 36 possible combinations are equally probable. That's why I made sure to note that the dice were fair, so that no one would suggest they were loaded in order to support their incorrect answer. :P

For example the numbers 2 and 12 can only be rolled by either rolling two 1's or two sixes. This is the only way to roll these numbers. The number 7 can be rolled six ways. 1/6, 2/5, 3/4, 6/1, 5/2, 4/3 giving it a better probability of being rolled.

You're confusing the totals of the dice with the 36 possible individual rolls. When you add the two numbers together, there are only 11 possible outcomes -- from 2 to 12.

I think your question could have been phrased better, Gord. This is your question.
Now here's the question: What are the odds that I rolled two 1s?

The odds on you rolling snakes eyes are one in six.

But I don't think this is the question you are asking?

You're partially correct, but mostly incorrect. First of all, when rolling two fair dice, the chances of rolling snake eyes is one in 36, because there is only one possible combination from the 36 equally-likely possible outcomes. Second of all, yes, I could just about always have better phrased any question I've ever asked ( :P ), but I still phrased it adequately for the brain teaser.
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Re: A roll of the dice

Postby landrew » Tue Nov 10, 2009 9:37 pm

I never understood the fascination with math. Everything adds up because it has to, end of story.
Let's move on.
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Re: A roll of the dice

Postby Gord » Tue Nov 10, 2009 9:59 pm

I agree, landrew, since you're not into math, you should definitely move on. :mrgreen:
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Re: A roll of the dice

Postby Gord » Tue Nov 10, 2009 10:21 pm

Now, as for the people still interested in math: Let's see if I can clear up the confusion with some simple diagrams.

First of all, when I roll two fair six-sided dice (I will colour the dice red and blue), the possible outcomes are shown in the following square:

1,1 . 1,2 . 1,3 . 1,4 . 1,5 . 1,6
2,1 . 2,2 . 2,3 . 2,4 . 2,5 . 2,6
3,1 . 3,2 . 3,3 . 3,4 . 3,5 . 3,6
4,1 . 4,2 . 4,3 . 4,4 . 4,5 . 4,6
5,1 . 5,2 . 5,3 . 5,4 . 5,5 . 5,6
6,1 . 6,2 . 6,3 . 6,4 . 6,5 . 6,6

These are all the possible outcomes, and, because the dice were fair, we know they are equally likely to have happened.

Next, when I tell you that I rolled at least one 1, you now know that 25 of these possible outcomes could not have happened; so you can discount them from the possibilities on the grid, which leave you with this:

1,1 . 1,2 . 1,3 . 1,4 . 1,5 . 1,6
2,1
3,1
4,1
5,1
6,1

As you can see, there are now eleven remaining possiblities. They are each still equally likely.

When I further ask for the odds that I rolled two 1s, you can look at the chart, find the one place where that occurred, and know the chances of that happening were 1 in 11.
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Re: A roll of the dice

Postby Gord » Tue Nov 10, 2009 10:22 pm

And in conclusion, I've allowed confusion to remain over the words "odds" and "chances." The chances of it happening are normally given as X in Y; the odds of it happening are normally given as X:(Y-X). So, the chances are 1 in 11, and the odds are 1:(11-1), or 1:10.
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Re: A roll of the dice

Postby Martin Brock » Wed Nov 11, 2009 3:51 am

Very nice.
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Re: A roll of the dice

Postby Gord » Wed Nov 11, 2009 4:12 am

Thanks. I hope that makes sense to everyone now.

There are some similar puzzles I could have posted, but I liked this one instead because it deals with dice which I use quite often. The other questions involve things like flipping coins or guessing which curtain conceals the best prize on The Price Is Right.
"Knowledge grows through infinite timelessness" -- the random fictional Deepak Chopra quote site
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Re: A roll of the dice

Postby vanderpoel » Wed Nov 11, 2009 12:05 pm

Gord wrote:Next, when I tell you that I rolled at least one 1, you now know that 25 of these possible outcomes could not have happened; so you can discount them from the possibilities on the grid, which leave you with this:

No, I know that 25 of these possible outcomes could have happened, past tense, just as your question refers to the past tense of the possibility of 2 1s occurring. Itʻs not a question of math, but a question of grammar. Nice graph though.
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