### Special game mode of throwing a fair dice; Use Sum of Geometric Sequence to find the probability of certain situation

Question Sample Titled 'Special game mode of throwing a fair dice; Use Sum of Geometric Sequence to find the probability of certain situation'

Zac and Danielle take turns to throw a fair die until one of them gets a number '${1}$' or '${5}$' . Zac throws the die first. Find the probability that Danielle gets a number '${5}$' .

A
$\dfrac{{1}}{{5}}$
B
$\dfrac{{3}}{{10}}$
C
$\dfrac{{1}}{{2}}$
D
$\dfrac{{1}}{{3}}$

 Req. Prob. $=\dfrac{{4}}{{6}}\cdot\dfrac{{1}}{{6}}+{\left(\dfrac{{4}}{{6}}\cdot\dfrac{{4}}{{6}}\cdot\dfrac{{4}}{{6}}\cdot\dfrac{{1}}{{6}}\right)}+{\left(\dfrac{{4}}{{6}}\cdot\dfrac{{4}}{{6}}\cdot\dfrac{{4}}{{6}}\cdot\dfrac{{4}}{{6}}\cdot\dfrac{{4}}{{6}}\cdot\dfrac{{1}}{{6}}\right)}+\ldots$

(The first term means: Zac did not get '${1}$' or '${5}$' in the first round, and Danielle get '${5}$' in the second round.

The second term means: Zac did not get '${1}$' or '${5}$' in the first round, and Danielle did not get '${1}$' or '${5}$' in the second round, and Zac did not get '${1}$' or '${5}$' in the third round, and Danielle get '${5}$' in the fourth round. The remaining terms are similar.)

 The formula forms a geometric sequence, having the first term $\dfrac{{4}}{{6}}\cdot\dfrac{{1}}{{6}}$ and ratio $\dfrac{{4}}{{6}}\cdot\dfrac{{4}}{{6}}$ . ∴   Req. Prob. $=\dfrac{{\dfrac{{4}}{{6}}\cdot\dfrac{{1}}{{6}}}}{{{1}-\dfrac{{4}}{{6}}\cdot\dfrac{{4}}{{6}}}}$ Sum of G.S.  $=\dfrac{{1}}{{5}}$

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