Dice probability of rolling 9 or more on 2 six sided dice w/ re-roll

[bjorn]

Hello!
I have a pretty simple problem with dice that is made really hard by a re roll.
The problem is this: I want to know the percentage probability of rolling 9 or more on 2 six sided dice (2d6) BUT i get to re roll one of the dice.
How do you even start to calculate this?

 

[tkhunny]

p(9 or more on the first roll) for starters.

It is reasonable to think that you would NOT reroll if this first roll produced 9 or more.

p(9 or more on fist roll) + p(8 or less on first roll) * p(fixing it with a reroll)

For p(fixng it with a reroll), simply specify the entire distribution:

What is the value of the die you are NOT rerolling?

1 - p(fixing it) = 0
2 - p(fixing it) = 0
3 - p(fiing it) = p(rolling 6) = 1/6
4 - Continue in this way...

Exploration: It my also be reasonable to assume you will reroll the die wit the lesser value on the first roll. Does this change the result?