A) \[\frac{{{3}^{50}}+{{2}^{50}}}{{{2.3}^{50}}}\]
B) \[\frac{{{3}^{50}}+1}{{{2.3}^{50}}}\]
C) \[\frac{{{3}^{50}}-1}{{{2.3}^{50}}}\]
D) \[\frac{{{3}^{50}}-{{2}^{50}}}{{{2.3}^{50}}}\]
Correct Answer: B
Solution :
[b] The coin can turn up heads 0, 2, 4, ..., 50 times to satisfy the condition. Hence probability is, \[P{{=}^{50}}{{C}_{0}}{{\left( \frac{2}{3} \right)}^{0}}{{\left( \frac{1}{3} \right)}^{50}}{{+}^{50}}{{C}_{2}}{{\left( \frac{2}{3} \right)}^{2}}{{\left( \frac{1}{3} \right)}^{48}}+....{{+}^{50}}{{C}_{50}}{{\left( \frac{2}{3} \right)}^{50}}{{\left( \frac{1}{3} \right)}^{0}}\] \[=\frac{{{\left( \frac{1}{3}+\frac{2}{3} \right)}^{50}}+{{\left( \frac{1}{3}-\frac{2}{3} \right)}^{50}}}{2}\] \[=\frac{{{3}^{50}}+1}{{{2.3}^{50}}}\]
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