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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) ?

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    $\begingroup$ My interpretation of the question is that you roll a die repeatedly until you see a "large number." Note that a die can land on the same values multiple times in a row. Here are some possible outcomes of the game: (1,3,1,5)... (4,2,6)... (5)... (1,6)... (4, 4, 2,4,1,1,1,1,1,5)... The question can be reworded then "What is the probability that having thrown three dice, they are all 'small' numbers?" What do you know about independent events? What do you know about multiplying probabilities? $\endgroup$
    – JMoravitz
    Commented Aug 30, 2016 at 7:34

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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.

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    $\begingroup$ I wouldn't get in involved with the chance that he stops at all, because then you've either got to calculate the prob of stopping on 1,2 or 3 and take that away from 1 - otherwise sum an infinite series where he stops on 4 or later. $\endgroup$
    – Cato
    Commented Aug 30, 2016 at 10:13
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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

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