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14. probability — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS14. probability1 Mark
MCQ
Two persons $A$ and $B$ throw a (fair)die (six-faced cube with faces numbered from $1$ to $6$ ) alternately, starting with $A$. The first person to get an outcome different from the previous one by the opponent wins. The probability that $B$ wins is
  • A
    $\frac{5}{6}$
  • ✓
    $\frac{6}{7}$
  • C
    $\frac{7}{8}$
  • D
    $\frac{8}{9}$
✓

Answer

Correct option: B.
$\frac{6}{7}$
b
(b)

Given, Probability of win $=\frac{5}{6}$

Probability of loss $=\frac{1}{6}$

Probability of $B$ wins =

$P(B)+P(\overline{B A} B)+P(\bar{B} \bar{A} \bar{B} \bar{A} B)+\ldots$

$=\frac{5}{6}+\frac{1}{6} \times \frac{1}{6} \times \frac{5}{6}+\frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} \times \frac{5}{6}+\ldots$

$=\frac{5}{6}\left(1+\frac{1}{6^2}+\frac{1}{6^4}+\frac{1}{6^6}+\ldots\right)$

$=\frac{5}{6}\left(\frac{1}{1-\frac{1}{36}}\right)=\frac{30}{35}=\frac{6}{7}$

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