If I say I am going to roll a die 4 times, the probability of getting 4 of the same number I picked in a row is 0.077%

Each time I roll the die, the probability of getting the number I picked is 16.67%

The probability of each roll achieving the result you want actually increases as the previous rolls are in your favour.

If you roll 2 of the same in a row, the odds of getting two more is 2.78%, and if you get 3 in a row, the odds of getting the final one is again 16.67%.