Suppose you are playing a dice game that goes as follows You

Suppose you are playing a dice game that goes as follows. Your goal id to roll a 1. On your first turn you get to roll a single six sided die. If you roll a 1, you win the game. If you do not, you add another die, so that you have two six sided dice. Then on your second turn you roll both die and try to get a 1. If you get at least one 1 you win. Otherwise you add another die and continue on. You continue to add dice one at a time until you roll a 1. On what turn is this game most likely to end?

Solution

If this game continues till n\'th roll (without showing a \'1\'), the next step will have (n+1) die.

P(rolling 1 in n+1 die) = 1 - P(no \'1\' in n+1 die) =1 - (1/6)^(n+1)

We note that this probability increases as n increases. There is no integer value n where this probability could attain maxima. Thus this game is most likely to end at infinity.