A) \[\left( \frac{{{5}^{n-1}}}{{{6}^{n-1}}} \right)+\left( \frac{{{5}^{n}}}{{{6}^{n}}} \right)\left( \frac{1}{6} \right)\]
B) \[\left( \frac{{{5}^{n}}}{{{6}^{n}}} \right)+\left( \frac{{{5}^{n-1}}}{{{6}^{n-1}}} \right)\left( \frac{1}{6} \right)\]
C) \[\left( \frac{{{5}^{n}}}{{{6}^{n}}} \right)+n\left( \frac{{{5}^{n-1}}}{{{6}^{n-1}}} \right)\left( \frac{1}{6} \right)\]
D) \[\left( \frac{{{5}^{n}}}{{{6}^{n}}} \right)+n\left( \frac{{{5}^{n-1}}}{{{6}^{n-1}}} \right)\left( \frac{1}{{{6}^{2}}} \right)\]
Correct Answer: D
Solution :
\[{{B}_{1}}{{B}_{2}}....{{B}_{n}}\] |
Case I Di never show 1 |
Probability \[={{\left( \frac{5}{6} \right)}^{n}}\] |
Case II \[{{D}_{2}}\]shows 1 (one time) then \[{{D}_{1}}\] |
Probability \[=\left\{ {}^{n}{{C}_{1}}{{\left( \frac{5}{6} \right)}^{n-1}}\left( \frac{1}{6} \right) \right\}\left( \frac{1}{6} \right)\] |
Total probability \[={{\left( \frac{5}{6} \right)}^{n}}+n\left( \frac{{{5}^{n-1}}}{{{6}^{n-1}}} \right)\left( \frac{1}{{{6}^{2}}} \right)\] |
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