Game of Dice

Probability Level 3

Lucas and Logan play a game of dice. Lucas goes first, rolls a single six-sided die, and wins if he rolls a two or lower. If Lucas doesn't win, Logan goes next, rolls a single six-sided die, and wins if he rolls a five or higher. If Logan doesn't win, Lucas goes again, then if Lucas doesn't win, Logan goes again, and so on, until one of them wins. What is Lucas's probability of winning the full game?

(Write your answer in decimal format)


The answer is 0.6.

Garam O'Brien by Garam O'Brien , 50, USA

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1 solution

∑ n = 0 ∞ 2 6 ( 4 6 ) 2 n = 2 6 ∑ n = 0 ∞ ( 4 6 ) 2 n = 2 6 . 1 1 − ( 4 6 ) 2 = 0.6 \sum_{n=0}^{\infty}\frac{2}{6}(\frac{4}{6})^{2n}=\frac{2}{6}\sum_{n=0}^{\infty}(\frac{4}{6})^{2n}=\frac{2}{6} . \frac{1}{1-(\frac{4}{6})^2}=0.6

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