Questions: Seven of the 100 digital video recorders (DVRs) in an inventory are known to be defective. What is the probability that a randomly selected DVR is defective? The probability is

Seven of the 100 digital video recorders (DVRs) in an inventory are known to be defective. What is the probability that a randomly selected DVR is defective? The probability is
Transcript text: Seven of the 100 digital video recorders (DVRs) in an inventory are known to be defective. What is the probability that a randomly se is defective? The probability is $\square$
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Solution

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Solution Steps

Step 1: Define the Problem

We are tasked with finding the probability that a randomly selected digital video recorder (DVR) from an inventory of 100 DVRs is defective. Out of these, 7 DVRs are known to be defective.

Step 2: Set Up the Hypergeometric Distribution

To solve this, we can use the hypergeometric distribution, which is defined as:

\[ P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}} \]

where:

  • \( N = 100 \) (total number of DVRs),
  • \( K = 7 \) (total number of defective DVRs),
  • \( n = 1 \) (number of DVRs drawn),
  • \( k = 1 \) (number of defective DVRs drawn).
Step 3: Calculate the Probability

Substituting the values into the formula, we have:

\[ P(X = 1) = \frac{\binom{7}{1} \binom{93}{0}}{\binom{100}{1}} \]

Calculating each component:

  • \( \binom{7}{1} = 7 \)
  • \( \binom{93}{0} = 1 \)
  • \( \binom{100}{1} = 100 \)

Thus, we can compute:

\[ P(X = 1) = \frac{7 \cdot 1}{100} = \frac{7}{100} = 0.07 \]

Final Answer

The probability that a randomly selected DVR is defective is

\[ \boxed{0.07} \]

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