Question

What is the chance of rolling a 4 with a single die three times in a row

Answer 1

The probability of rolling 4 in one roll is
`(1)/(6)`.
So the probability of rolling 4 three times in a row is
`(1)/(6)\cdot(1)/(6)\cdot(1)/(6)=(1)/(216)`

Answer 2

The key thing to remember when you're working a problem like this is that
the cube has no memory ... it doesn't know what happened on the last roll,
so the chance of any particular thing happening on the 2nd roll, or on the
100th roll, is the same as it was on the first roll.

Probability of success = (number of ways to succeed) / (total possibilities)

Number of ways for a 4 to come up . . 1
Total possible results of the roll . . . . . 6

Probability of rolling a 4 . . . . . 1 / 6 .

Probability of rolling two 4s . . . (1/6) x (1/6)

Probability of rolling three 4s . . (1/6) x (1/6) x (1/6) = 1/216 = about 0.46% .