Multi-roll, 12 sided die outcomes (used brute force 725,000 die rolls)

[sjelso]

I have forgotten my grad school probability and need some help with the following.

I am trying to create a .xls that I can use to look at the odds of multiple dice throws.

Quantity of dice = 10 (n)
# of sides per dice = 20 (s)

Each dice has the following colors instead of numbers:

4 Yellow sides (this could be 1,2,3,4)
3 Blue sides (5,6,7)
2 Green sides (8,9)
1 Red side (10)
1 White side (11)
1 Black side (12)

I would like to understand/know the formula(s) that would help determine the probability of the following different scenarios:

Scenario (each scenario a separate 10 dice roll):

1) Probability of rolling a combination of yellows across the 10 dice, lets say 12 yellow side results
2) Probability of rolling 2 Red sided across the 10 dice - is this 2 times (P(X=r) = nCr * pʳ * (1-p)ⁿ⁻ʳ)?
3) Probability of rolling 3 yellow, 4 blue, 1 green, 0 red, 1 white and 3 black

I have looked at some joint probability examples and have tried to brute force the answer in .xls but I am not quite getting it.

Thank you in advance for any assistance....Steve

 

[staassis]

The set of your questions is big enough. It cannot be covered with morning air transparency in just 1-3 paragraphs. You will be able to answer all these questions if you read chapters 1-3 of

Ross, S. M. (2012). A First Course in Probability (9th ed). Pearson Education Limited.

They are quite short... Alternatively, if mathematics is not your favorite time spending, you can Monte Carlo simulate all the probabilities.

 

[sjelso]

The set of your questions is big enough. It cannot be covered with morning air transparency in just 1-3 paragraphs. You will be able to answer all these questions if you read chapters 1-3 of

Ross, S. M. (2012). A First Course in Probability (9th ed). Pearson Education Limited.

They are quite short... Alternatively, if mathematics is not your favorite time spending, you can Monte Carlo simulate all the probabilities.

Thank you, I found a .pdf of the book online and will read through the first 3 chapters.

 

[staassis]

Great. I suspect the book will prove helpful with many other mathematical problems that you will face. The concepts presented in chapters 1-3 are fundamental. The book is a relatively easy read. I like all of Ross's books, in fact.

 

[sjelso]

I ended up using (maybe it is Monte Carlo...or close) brute force to arrive at a probability table with 750,000 rolls of ten dice (RND).

I believe I got it correct. The top table is the sum of the Exact Hits by color / 725,000 rolls of 10, 12 sided dice.

So in line Y: The odds of rolling just 1 yellow hit (4 yellows per 12 sides) = 8.7%. That is low b/c it is not that easy to roll just one yellow when there are 4 per die.

Somewhat counter intuitive, for me at least..at first, was that 3 blues (3 per 12 sides) is more likely than 3 or 4 yellows. 28.2% vs 26.0%.

The criteria section is to determine the likelihood of rolling a certain quantity of each color in a roll. It seems that even what one may think is a common role 3Y, 3B, 2G, 1R, 1W, 0B is actually very rare. Just 0.56% chance of that specific roll (4,005 matches / 725,000)....If I got it all correct.

A match is when the Total (TTL) = 6 (no matches in the first 15 rows shown below).

I would not object if someone wants to give this a once-over to see if it holds water.

Cheers..Steve

(still going to do my probability reading too)