# How to calculate the probability of reaching a certain number when rolling a six allows you to re-roll and add to sum?

In the Shadowrun RPG (at least, the edition I remember) one of the core mechanics of the game was that you rolled a number of dice equal to your skill against a difficulty number. Rolling a 6 meant you could re-roll that die and add the result to the total.

So, for example, say I have a skill of 3 and that I have to roll against a 10:

Die 1 = 4 -> no luck
Die 2 = 6. I re-roll a 2 for a total of (6+2=) 8 -> still no luck
Die 3 = 6. I re-roll for another 6. I re-roll again and get a 3. Total = 6+6+3 = 15 -> Yay!


How can I calculate the probability of me succeeding, knowing only the number of dice I have and the target number?

• What is your strategy for deciding when to re-roll? Will you always re-roll individual sixes? Are you aiming to exceed the "target number"? To equal or exceed it? When you re-roll a six and get another six, can you re-roll that (ad infinitum)? – whuber Mar 9 '17 at 15:58
• You always re-roll sixes until you have reached or exceeded the target. Equalling the target also counts as success. You can reroll as infinitum (but you'd reach a target number eventually of course). – LaundroMat Mar 9 '17 at 18:23
• I believe the thread at stats.stackexchange.com/questions/41238 might already hold the answers you are looking for. – whuber Mar 9 '17 at 18:38