Not sure I remember all the ways you can end up in jail, but....
Way 1] The probability of rolling doubles on a turn is 1/6. The probability of rolling three consecutive doubles is therefore (1/6)³ ~ 0.46%.
Way 2] Landing on "Go To Jail" which is 30 spaces from the start. Can be done w/o rolling a third double first by rolling (12,12,6) or (12,10,8) or (10,12,8) or (10,10,10)
= 3*(1/36)²*(4/36) + (1/36)²*(2/36) ~ 0.03%
So far that's a total probability of 0.49%
Way 3] Landing on a "Community Chest" or "Chance" square w/o first rolling three doubles nor rolling exactly 30 and then drawing a "Go To Jail" Card.
Chances from first Chance/Community Chest
= (1/36)*(1/16) + (1/36)*(4/36)*(1/16) + (1/36)*(1/36)*(2/36)*(1/16) ~ 0.19%
Up to 0.68% total probability now
Chances from second Community Chest
= [2*(1/36)²*(2/36)*(15/16) + 2*(1/36)²*(4/36)*(15/16) + (1/36)²*(6/36)*(15/16) + 2*(1/36)²*(2/36)* + 2*(1/36)²*(4/36) + (1/36)²*(6/36) + (1/36)*(2/36) + (1/36)²*(2/36) + 2*(1/36)²*(4/36) + (1/36)²*(6/36) + (1/36)*(4/36) + (1/36)²*(2/36) + (1/36)²*(4/36) + (1/36)²*(6/36) + (1/36)*(6/36) + (1/36)²*(2/36) + (1/36)²*(4/36) + (1/36)*(4/36) + (1/36)²*(2/36)] * [1/6]
~ .09%
Up to 0.77% total probability now...
Needless to say accounting for all possibilities of doubles/ways to roll particular numbers and evading previously drawn chance/community chest cards gets out of hand fast and I will stop my precise calculation here at the second community chest space. My total estimate is definitely less than 1%, maybe around 0.85% or so if I extrapolate how my subsequent calculations were decreasing in likelyhood. Note, you can't get to the last chance square before GO w/o rolling three doubles first. On the order of 1 out of every 120 times you start the game. More frequent than I expected actually.
