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Standard XII

Mathematics

Total Probability Theorem

A distributor...

Question

A distributor of bean seeds determines from extensive tests that $5$ % of seeds will not germinate. He sells the seeds in packets of $200$ and guarantees $90$ % germination. The probability that a particular packet will violate the guarantee is

\frac{e^{- 10} (10)^{10}}{10!}

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\frac{e^{- 10} (10)^{20}}{10!}

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1 - 20 \sum x = 0 \frac{e^{- 10} (10)^{x}}{x!}

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1 - \infty \sum x = 0 \frac{e^{- 10} (10)^{x}}{x!}

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Solution

The correct option is C $1 - 20 \sum x = 0 \frac{e^{- 10} (10)^{x}}{x!}$
The probabilityof a seed not germinating $= p = \frac{5}{100} = 0.05$
$λ =$ mean number of seeds in a sample of 200. which do not germinate $= n p = 200 \times 0.05 = 10$
Let X = R.V. = number of seeds that do not germinate
A packet will violate guarantee if it contains more than 20 non-germinating seeds
Probability that the guarantee is violated
$= P (X > 20) = 1 - P (X \leq 20) = 1 - 20 \sum x = 0 \frac{e^{- 10} 10^{x}}{x!}$

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A distributor of bean seeds determines from extensive tests that 5% of seeds will not germinate. He sells the seeds in packets of 200 and guarantees 90% germination. The probability that a particular packet will violate the guarantee is

A distributor of bean seeds determines from extensive tests that $5$ % of seeds will not germinate. He sells the seeds in packets of $200$ and guarantees $90$ % germination. The probability that a particular packet will violate the guarantee is