Question

probability question:

choose 19 integers \[ 0 le X_r le 10 \]

find the probability their sum is greater than 120

Answer 1

\[0 \le X_r \le 10\]

Answer 2

i have an answer but i got it through a fairly brute force method

Answer 3

i would be really interested if there was a nice/general solution

Answer 4

hmm... maybe to find out the formula would be possible to think like that:
First consider groups of 19 integers, all equal that do summ more than 120:
1010101010....
9999...
88888...
77777....
Later for each of this group there would be a margin that the have over 120:
19*10-120=70
19*9-120=51
....
this margin could be summed by numbers smaller , but all equal, then the original number. This way you also will get groups of equal numbers with new margins.
I think this way you can manage to get all possible combinations and work out recursive type formula for the general solution.

Answer 5

this is the manner in which i proceeded, which might also be a possible foot-in to a formula (notice pascal triangle diagonals in rows)

each column is an extra integer added, each row shows the probability of the sum being
1,2,3,4,5,... etc

Answer 6

looks promising, :)

Answer 7

Is the answer around 3.8%?

Answer 8

0.031975056

Answer 9

this was the original problem this came from:

Consider a computer game where you play a level 1 character with 150 HP.
Each time your character levels, you have a random chance of gaining between 20 to 30 HP (So you can be level 2 with a HP anywhere between 170 to 180.
At level 20:

given that HP is discrete (takes integer values)

a) Calculate the range of HP your character could have
b) Calculate the probability your character has over 650 HP