Who knows probability?

[toMACsh]

Playing a new dice game.

Ok, now discounting the fact that a die is unbalanced (because a side has 6 pips and another only has 1) what are the chances of scoring in this scenario:

Only a 1 (100) and a 5 (50) will score points (for starters anyway)

For a single die, the odds are 1:3 of scoring.
You have to score to continue and add to your running total.

You have 6 dice. What are the odds of getting at least one scoring die? Well, I'd say 1:6 but for each one individually, it's that 1:3 chance of getting either a 1 or a 5. So it must be better than 1:6 ,right?
Now you set that aside and roll 5 dice. What are the odds now?
4 dice
3 dice
2 dice
back to one with a 1:3 chance

Can anyone fill in 2 through 6?

 

[iggibar]

The probability depends on whether or not independency is a factor. If it is dependent, then your probability factor would include the probability times the number of dice. If not dependent, then the probability would remain a constant 1/3 for every new dice rolled.

I said this because this:

You have 6 dice. What are the odds of getting at least one scoring die? Well, I'd say 1:6 but for each one individually, it's that 1:3 chance of getting either a 1 or a 5. So it must be better than 1:6 ,right?
Now you set that aside and roll 5 dice. What are the odds now?

makes it seem like they are not dependent.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

I'm on vacation, so I don't have my trusty Ti-89 with me, but I know that there are a total of
46656 combinations from 6 dice(6x6x6x6x6x6),
7776 combinations from 5 dice(6x6x6x6x6),
1296 combinations from 4 dice(6x6x6x6),
216 combinations from 3 dice(6x6x6),
36 combinations from 2 dice(6x6),
and 6 combos from 1 die(1x6).

What I like to do is figure out the chances of not rolling that combo, then subtract it from the total combinations available. This comes in handy in the future statistics classes.

1. Chances of rolling a 1 or 5 from 1 die = 2/6. Easy one. lol

2. Chances of not rolling them with 2 dice = 4x4=16 out of 36.
36-16= 20/36 or 55.5% probability of getting a 1 or 5 from 2 dice.

3. Chances of not rolling them with 3 dice = 4x4x4=64 out of 216 combos.
216-64= 152/216 or 70.3%probability of getting a 1 or 5 from 3 dice.

4. Chances of not rolling them with 4 dice = 4x4x4x4=256 out of 1296 combos.
1296-256= 1040/1296 or 80.2% probability of getting 1 or 5 from 4 dice.

5. Chances of not rolling them with 5 dice = 4x4x4x4x4=1024 out of 7776 combos.
7776-1024= 6752/7776 or 86.8% probability of getting a 1 or 5 from 5 dice.

6. Chances of not rolling them with 6 dice = 4x4x4x4x4x4=4096 out of 46656 combos.
46656-4096= 42600/46656 or 91.3%probability of getting a 1 or 5 from 6 dice.


PS: Take what I wrote up there with the knowledge that I am thoroughly enjoying this drinkful(new word!) day

 

[toMACsh]

Ok, what determines dependency? And how does that impact the probabilities?

I get how your calculations work. I assume the chance does not change whether you roll all six dice at the same time, or if you choose to start your turn by rolling one die six times (6 different dice, or the same one - doesn't matter). I further assume that your probabilities include instances of getting 5 scoring dice on the first roll, 4 out of 5, etc. But of course, you only need one scoring die to continue your turn in the game.

I figure that if I roll one die once, I get that 1:3 chance. But if I roll that one die over and over, eventually the 1 or 5 result should come up 33.3% of the time, right?

Now, is there a difference between probability and possibility? That is to say it might be possible to roll a single die repeatedly and NEVER get the desired result, a 1 or a 5, but it's not probable. In light of that, is it correct or incorrect to say that it would be impossible to NEVER get the desired result, or is it merely highly unlikely? We're entering the realm of theory here now...

There are also points awarded for getting a run on the first roll (1,2,3,4,5, and 6) or six of a kind. But there is only one chance of getting six 1s, and one for six 2s, etc. Thus there should be 6 chances out of the 46,656 combinations to score in that category. You get 3,0000 points for that. However, you only get 1,500 points for the 1 thru 6 result. There must be at least 36 chances of getting that, since each of the dice can be any number. So I guess they have their points awarded logically in this case.

However, a 1 being 100 points and a 5 being 50 makes no sense. Further, they award
600 for 3 sixes
500 for 3 fives
400 for 4 fours
300 for 3 threes
200 for 3 twos
and, wait for it
300 for 3 ones!

The chance of getting any one of those should be the same, but the points are different! I guess they have to give you 300 instead of 100 for the three 1s because they're giving you 100 for one 1.

 

[Slydude]

A pox on both your houses for giving me an enormous headache. lol. I have not thought about this kind of stuff since high school many years ago.

 

[dauber]

Well, who's to say that the pips weren't systematically designed so that all six sides are balanced???

 

[iggibar]

A 6-sided die is technically balanced because the pips are not removed, but are either pushed into the die, or are cast around the pips when formed- therefore each side retains its consistent weight compared to a solid cube. This does not even include the fact that gravity and beliefs have a greater effect on what lands up than the actual weight of the sides.

Independency is determined by whomever is explaining the rules.

 

[iggibar]

I further assume that your probabilities include instances of getting 5 scoring dice on the first roll, 4 out of 5, etc. But of course, you only need one scoring die to continue your turn in the game.

My calculations include getting either a single 1 or 5 from any number of dice rolled at once. (dependent)

I figure that if I roll one die once, I get that 1:3 chance. But if I roll that one die over and over, eventually the 1 or 5 result should come up 33.3% of the time, right?

That's where independency comes into play. If you role one die and the outcome does not factor into the next roll, then it's independent, meaning the statistic of rolling a 1 or 5 will remain the same for every roll. If it's dependent, the chances of getting a 1 or 5 get multiplied by the total number of dice rolled at that time. (meaning the probability of rolling a 1 or 5, from 5 dice at once, is increased).

Now, is there a difference between probability and possibility...?

Yes.
Possibility refers to something that has a greater than 0.01% chance of happening...which infers that it is guaranteed to happen, regardless of how minuscule the chance of it is.
One way to remember this is from the actual word= possi-bility (ability for it to be possible).

Probability refers to how likely it is to happen, through statistical observations, which can include 0. Also, in statistics, you never claim something is "impossible," but you either fail to reject it, or accept the null hypothesis. It describes how likely it is for an event to occur. Probability is always viewed as a proportion.

The probability of rolling a #2 side of a die is 6:1.
The possibility of rolling a #2 side of a die is viewed as a plausible measure, which leads to answering with "We fail to reject the hypothesis that rolling a #2 has a different outcome of 6:1." Which basically is a mathematical nerd's way of saying "Yes, it will happen."

With that said though, the word "possibility" is frowned upon to math professors who "correctly" teach Statistics. One reason for this is that humans have a limit as to what they can perceive as possible, and cannot indefinitely measure it to come up with an answer, which precise measurements is something math and science require to one or other degree.

 

[toMACsh]

Well, who's to say that the pips weren't systematically designed so that all six sides are balanced???

Gladys Knight?

 

[RavingMac]

gladys knight?

rofl

 

[_Stormin]

Gladys Knight?

You have won the Internet for the day sir...

 

[toMACsh]

A 6-sided die is technically balanced because the pips are not removed, but are either pushed into the die, or are cast around the pips when formed- therefore each side retains its consistent weight compared to a solid cube.

Well, the black paint on the side with 6 pips certainly weighs more than that on the side with one.

 

[BrianLachoreVPI]

Well, the black paint on the side with 6 pips certainly weighs more than that on the side with one.

Not on precision dice. The pips are from material of the same density as the dice so the volume of the material remains the same - and the weight is distributed equally.

 

[toMACsh]

I don't imagine that dice which come in a parlor game are precision dice. I think what you're mentioning are dice that have no "dimples" but rather the pips are flush with the side. These dice have dimples and paint, which will eventually wear off. Other dice I have just have painted on pips, no dimples. That paint wears off even faster.

 

[BrianLachoreVPI]

True - I guess I made an assumption that a conversation centered around determining probability would have a stable starting point. Makes it a little trickier to be completely accurate if the dice aren't 'fair' dice.

Gladys Knight?

LOLOL

 

[Kevriano]

This thread is so confusing it doesn't even fit on my laptops screen without me scrolling across.
Probability? It is very probable that I will never understand what on earth you are talking about

 

[toMACsh]

True - I guess I made an assumption that a conversation centered around determining probability would have a stable starting point. Makes it a little trickier to be completely accurate if the dice aren't 'fair' dice.

Yeah, that's why my first post said this:

Playing a new dice game.

Ok, now discounting the fact that a die is unbalanced (because a side has 6 pips and another only has 1) what are the chances of scoring in this scenario:

This thread is so confusing it doesn't even fit on my laptops screen without me scrolling across.

I noticed it being very wide too. No images. Not sure why it got "fat".

Back to my game....
You get lots of points for 2 triplets (2 sets of three) I guess there are only 30 such combinations since the other six that would seem to qualify are 6 of a kind, a whole different category.

You also get a big score for three pairs. Again, if two of those pairs are the same number, that's a different category which I call a "full house". But I'm not sure how to figure out how many of those there are. 2 ones, then 2 twos, then two threes through sixes, etc. iggi, a little help here

 

[McBie]

I started thinking about all of this on Christmas eve and I still haven't figured it out ... please wait .... processing....

Cheers ... McBie