KCC's Quizzes AQQ287 about throwing 3 dice

1. When you throw 3 dice, max probability is for S=10, 11. You can see this by calculating all the occurrences or by using excel or (better) noting that the average value (expectation value) is 3x (avg outcome 1 dice) = 3x 3.5=10.5 (in the limit of high number of rolls). Since you cannot get a fractional number, max probability will be for the closest integers 10 and 11. The avg outcome of 1 dice is (1+2+3+4+5+6)/6=3.5.

2. Similarly, for n dice you expect the average outcome= 3.5*n being the most probable outcome.

Yes, I agree with your inputs! The relation 3.5*n with n the number of dice can be easily verified for n up to 3. The only pending question is what is the probability (thus the number of times the 3.5 * n result will appear. As I mentioned earlier: I don't have that answer...

You have a a discrete probability distribution so you need to calculate the "positive" number of cases and divide it by the total number of cases. For n dice, total number of cases is 6ⁿ. Difficult part is to calculate the number of positive cases. Be G(x)= x¹ + x² + x³ +x⁴ +x⁵ +x⁶ the polynomial function associated with a single dice. If you have n dice, the total number of ways you can get 3.5*n is the coefficient of x^3.5*n  when you, or most probably an algorithm or a tool like wolfram alpha, expand [G(x)]ⁿ (let's assume 3.5*n is an integer, otherwise we follow same process for the closest integers).

For example, suppose we want to know the probability we get S=35 by rolling 10 dice. By letting wolfram alpha to expand [G(x)]10 , you get 4395256 as a coefficient for x³⁵. So the probability to get S=35 by rolling 10 dice is P(S=35)= 4395256 / 6¹⁰ = 0.0727 or something like 7.3%. 

The max probability will depend on n. I calculated the probability to get S=3.5*n for n=20 and the result is about 5.2%. On the other hand, the probability to get 3.5*n with n=30 is about 4.2%. So the higher the value of n, the lower the max probability, as you might expect by considering that  when n grows, you can get much more different sum results.   

Wow, I am impressed gpiarino , I was totally ignorant about the wolfram alpha technique. Will dig on it when during my summer vacation... Anywau, many thanks! 

Wow, I am impressed gpiarino , I was totally ignorant about the wolfram alpha technique. Will dig on it when during my summer vacation... Anywau, many thanks!