What is the probability of rolling a 1 exactly 3 times?

Question

You roll a number cube 4 times. What is the probability of rolling a 1 exactly 3 times?

Answer 1

you roll the dice 4 times

the first time you get a 1,2,3,4,5, or 6

the second time you roll the dice you get a 1,2,3,4,5, or 6 where each number is paired with one

of the first 6 numbers for 36 combinations

roll the dice a third time and the same six numbers (each one) is paired with the 36 combinations

for a total of 36*6=216 combinations

roll the dice a fourth time and these six numbers are each paired with the 216 combinations for a total of

6*216=1296 combinations

how many of these 1296 combinations are favorable ? ( meaning how many yield exactly three ones ?)

the favorable outcomes are 1,1,1, n where n=2,3,4,5,or 6 for a total of

5 favorable outcomes, or you could have...

1,1,n,1 where n=2,3,4,5,or 6 for a total of 5 more favorable outcomes, or you could have...

1,n,1,1 where n=2,3,4,5, or 6 for 5 more favorable outcomes, or you could have...

n,1,1,1 where n=2,3,4,5, or 6 for 5 more favorable outcomes for a total of 20 favorable outcomes out of a possible 1296 combinations

the probability would be 20/1296=5/324

 

 

Answer 2

Oops, in my first answer I missed the part about 4 rolls.  The corrected answer is below.  Thanks to Pierce O. for catching my mistake.

 

On a six-sides cube, the probability of rolling a one is 1 out of 6, or 1/6.  The probability of not rolling a 6 is 5/6.  So the probability P of rolling three  ones in four attempts is (or any number 3 consecutive times) is:

 

P = (1/6)³*(5/6) = 5/1296

 

However, there are 4 ways that you can roll 3 ones out 4 rolls:

 

1   1   1   not1

1   1  not1    1

1  not1   1   1

not1  1   1   1

 

So the probability is (5/1296)*4 = 5/324