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

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

## Comments

_{4}C_{3}* (1/6)^{3}^{3}^{3}_{4}C_{3}* (1/6)^{3}* (5/6)^{1}= 4! / ( (4 - 3)! * 3! ) * (1/6)

^{3}* (5/6)= 4 * (1/6)

^{3}* (5/6)= 4 * 1 / 216 * 5 / 6

= 5 / 324