Game of Dice and probability of playing

Question

You play a game of dice where you roll a fair 6-sided die, continue playing if you roll a 4 or lower, and stop if you roll a 5 or higher. What's the probability that you're still playing after 3 rolls?

Is this the right way to approach the game?

When i roll a 1 then i can roll the dice for 2,3,4 when i roll a 2 then i can roll the dice for 1,3,4 when i roll a 3 then i can roll the dice for 1,2,4 when i roll a 4 then i can roll the dice for 1,2,3

the probability of still playing after 3 rolls is 1/6(1+2+3+4) ?

Answer 1

Probability of stopping at a single throw $=2/6$ since you stop when you get $5$ or $6$.

Probability of stopping at exactly two throws is $\left(\frac{4}{6}\right)\left(\frac{2}{6} \right)$ by independentassumption.

Probability of stopping at exactly three throws is $\left(\frac{4}{6}\right)^2\left(\frac{2}{6} \right)$

Are you able to take it from here? check out geometric distribution.

Answer 2

The easiest way to do it is to calculate the probability that you DID NOT stop rolling on rolls 1,2 and 3 since on each roll there is a 2/3 chance of NOT stopping, that is to say

(2/3) x (2/3) x (2/3) = 8 / 27

you don't need to work out the probability that you DO stop at any time, you might notice that in a large numbers of players playing the game, 2/3 of them would carry on after each roll on average