In: Statistics and Probability

What is the probability of rolling 6, 6 sided die and achieving a score of 18? (6 die, each one is 6 sided)

On rolling 6 dices with 6 sides on each of them, what is the probability of getting a total sum of 18 across the 6 dices?

There are 6^{6} = 46656 different outcomes on rolling
six 6-sided die.

Doing this manually like we would do for 1 roll or 2 rolls or even 3 rolls is not an option here as this is infeasible and would take too much time for all practical purposes. The purpose of the question is not to test that either. Therefore, getting a little help from look up tables online -->

This table tells us the probability of getting said number (row) on said rolls of die (column).

For example, Roll 7 and 2 dice would mean the prob. of getting a 7 on rolling 2 six-sided die, and that is 6/36 = (1,6) (2,5) (3,4) (4,3) (5,2) (6,1)

Now to use this table to figure out the number of ways to get a 18 on rolling 6 die :

Focus on the column of 5 rolls.

There are 305 ways to get a 12 on rolling 5 die.

Now, what if we roll one more die and get a 6?

That would correspond to getting an 18 on rolling 6 die which is what we want!

Thus, we can list the possible outcomes this way by building up from the roll on 5 die.

a) 12 on 5 die + 6 on sixth die= 305 ways

b) 13 on 5 die + 5 on sixth die = 420 ways

c) 14 on 5 die + 4 on sixth die = 540 ways

d) 15 on 5 die + 3 on sixth die = 651 ways

e) 16 on 5 die + 2 on sixth die = 735 ways

f) 17 on 5 die + 1 on sixth die = 780 ways

These are all the different possible ways we can get an 18 on rolling 6 six-sided die :

Sum of all the ways gives us : 3431 ways

Thus, probability of getting an 18 on rolling 6 six-sided die :
3431 / 46656 = **0.0736**

Cheers!

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