Q81 / 160 · AP EAMCET 2018 (22 Apr Shift 2) · Mathematics

A manufacturer of locks knows that $2 \%$ of his product is defective. If he sells the locks in boxes each with 100 locks and guarantees that not more than 2 locks will be defective in a box, then the probability that a box will fail to meet the guaranteed quality is

  • Correct answer
    A
    $1-5 e^{-2}$
  • B
    $\sum_{k=2}^{100}{ }^{100} C_k\left(\frac{1}{50}\right)^k\left(\frac{49}{50}\right)^{100-k}$
  • C
    0.02
  • D
    $1-3 e^{-2}$

Solution

The random variable is the number of defective locks with mean $m=100 \times \frac{2}{100}=2$ So, the required probability $$ \begin{aligned} & =1-[P(r=0)+P(r=1)+P(r=2)] \\ & =1-\left[e^{-2} \frac{2^{\circ}}{0 !}+e^{-2} \frac{2^1}{1 !}+e^{-2} \frac{2^2}{2 !}\right]=1-5 e^{-2} \end{aligned} $$

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

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