What is the probability of getting an AVERAGE of six when you average the rolls of 4 dice?

Answer 1

Assuming the dice are unbiased standard dice, then

To get a mean average of 6, every dice must show a six → probability = 1/(6^4) = 1/1296 ≈ 0.000772

To get a median average of 6, then only 3 of the dice need show a six. In this case there are:

all 4 dice show a 6 = 1 way

To select 3 dice to show 6 leaves 1 die which can show any number 1-5. There are 4 ways this single die can be left giving 5 × 4 possible ways = 20 ways

This gives a total of 1 + 20 = 21 ways of it happening and the probability is 21/1296 ≈ 0.0162

To get a mode average of 6 is slightly complicated: all 4 dice can show a 6, three dice can show a 6 ,or 2 dice can show a 6 AND the other 2 dice show different numbers. There are:

1 way all 4 show a 6 (as above)

20 ways exactly 3 show a 6 (as above)

2 show a six:

2 dice are selectable in 4 × 3 = 12 ways.

Of the other 2 dice one can show one of 5 numbers and the other one of the remaining 4 which is 5*4 = 20, making 12*20 = 240 ways

Thus there are a total of 240+20+1 = 261 ways of it happening and the probability is 261/1296 ≈ 0.2014

Answer 2

It is 1/6^4 = 1/1296 = 0.0007716 (approx).

Answer 3

Please note that you can only get an average of 6 if each and every die gives you a six.