If 3% of the electric bulbs manufactured by a company are defective find the probability that
in a sample of 100 bulbs exactly 5 bulbs are defective.

To find the probability that exactly 5 bulbs are defective in a sample of 100, we can use the binomial probability formula.

The binomial probability formula is:

\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}
\]

Where:
- \( P(X = k) \) is the probability of having exactly \( k \) successes (defective bulbs) in \( n \) trials (total bulbs).
- \( n = 100 \) (the number of bulbs in the sample).
- \( k = 5 \) (the number of defective bulbs).
- \( p = 0.03 \) (the probability of a single bulb being defective).
- \( 1-p = 0.97 \) (the probability of a single bulb not being defective).

First, we calculate the binomial coefficient \( \binom{100}{5} \), and then use the formula:

\[
P(X = 5) = \binom{100}{5} \cdot (0.03)^5 \cdot (0.97)^{95}
\]

Now let's calculate this.The probability that exactly 5 bulbs out of 100 are defective is approximately \(0.101\), or 10.13%.

Would you like more details on this calculation or have any questions?

Here are some related questions to consider:
1. How would the probability change if the sample size were increased to 200 bulbs?
2. What is the probability that at least one bulb is defective in the sample of 100 bulbs?
3. How does the probability change if the defect rate increases to 5%?
4. What is the expected number of defective bulbs in the sample?
5. How would you calculate the probability that more than 10 bulbs are defective?

**Tip:** Understanding the binomial distribution is crucial when dealing with situations involving independent trials, such as manufacturing defects.

Calculate the probability that exactly 5 out of 100 electric bulbs are defective. Use the binomial probability formula with a defect rate of 3% per bulb. Get step-by-step solution and related questions answered.

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