Q260 / 270 · NDA 2017 (Phase 2) · Mathematics

The probability that a ship safely reaches a port is $\frac{1}{3}$. The probability that out of 5 ships, at least 4 ships would arrive safely is

  • A
    $\frac{1}{243}$
  • B
    $\frac{10}{243}$
  • Correct answer
    C
    $\frac{11}{243}$
  • D
    $\frac{13}{243}$

Solution

Probability that a ship reaches port $=\frac{1}{3}$. Probability that a ship not reaching port $=1-\frac{1}{3}=\frac{2}{3}$ Number of ships $(\mathrm{n})=5$. $r=4,5$ $\begin{aligned} & \mathrm{p}(4)+\mathrm{p}(5)={ }^5 \mathrm{c}_4\left(\frac{1}{3}\right)^4\left(\frac{2}{3}\right)^{5-4}+{ }^5 \mathrm{c}_5\left(\frac{1}{3}\right)^5\left(\frac{2}{3}\right)^{5-5} \\ & =5\left(\frac{2}{3^5}\right)+\frac{1}{3^5}\end{aligned}$ $=\frac{10}{243}+\frac{1}{243}=\frac{11}{243}$

Hence, the correct answer is option: 'C'.

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