Q. 1 The Probabilify of getting an item deflective
is 0.005 . What is the probabicity trat exactly 3 itmy
in a sample of 200 are deffective? (Given:

Ask by Santiago Schwartz. in India

Dec 28,2024

Question

Q. 1 The Probabilify of getting an item deflective is 0.005 . What is the probabicity trat exactly 3 itmy in a sample of 200 are deffective? (Given: $$ \left.e^{-1}=0.3679\right) $$

UpStudy AI · Accepted Answer

To find the probability that exactly 3 items in a sample of 200 are defective, we can use the binomial probability formula.

Given:
- Probability of getting a defective item = 0.005
- Sample size = 200
- Number of defective items = 3

The binomial probability formula is:
$$ P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k} $$

Where:
- $$ P(X = k) $$ is the probability of getting exactly k defective items
- $$ \binom{n}{k} $$ is the binomial coefficient, which represents the number of ways to choose k items from a sample of n
- $$ p $$ is the probability of getting a defective item
- $$ n $$ is the sample size
- $$ k $$ is the number of defective items

Substitute the given values into the formula:
$$ P(X = 3) = \binom{200}{3} \times 0.005^3 \times (1-0.005)^{200-3} $$

Now, we can calculate the probability using the binomial probability formula.
Calculate the value by following steps:
- step0: Calculate:
  $$ { }_{200}C_{3}\times 0.005^{3}\left(1-0.005\right)^{200-3}$$
- step1: Subtract the numbers:
  $$ { }_{200}C_{3}\times 0.005^{3}\times 0.995^{200-3}$$
- step2: Subtract the numbers:
  $$ { }_{200}C_{3}\times 0.005^{3}\times 0.995^{197}$$
- step3: Expand the expression:
  $$\frac{200!}{3!\times \left(200-3\right)!}\times 0.005^{3}\times 0.995^{197}$$
- step4: Subtract the numbers:
  $$\frac{200!}{3!\times 197!}\times 0.005^{3}\times 0.995^{197}$$
- step5: Reduce the fraction:
  $$\frac{200\times 199\times 198}{3!}\times 0.005^{3}\times 0.995^{197}$$
- step6: Multiply the terms:
  $$\frac{7880400}{3!}\times 0.005^{3}\times 0.995^{197}$$
- step7: Divide the terms:
  $$1313400\times 0.005^{3}\times 0.995^{197}$$
- step8: Convert the expressions:
  $$1313400\left(\frac{1}{200}\right)^{3}\times 0.995^{197}$$
- step9: Convert the expressions:
  $$1313400\left(\frac{1}{200}\right)^{3}\left(\frac{199}{200}\right)^{197}$$
- step10: Simplify:
  $$1313400\times \frac{1}{200^{3}}\times \left(\frac{199}{200}\right)^{197}$$
- step11: Multiply the terms:
  $$\frac{6567}{200^{2}}\times \left(\frac{199}{200}\right)^{197}$$
- step12: Evaluate the power:
  $$\frac{6567}{200^{2}}\times \frac{199^{197}}{200^{197}}$$
- step13: Multiply the fractions:
  $$\frac{6567\times 199^{197}}{200^{2}\times 200^{197}}$$
- step14: Multiply:
  $$\frac{6567\times 199^{197}}{200^{199}}$$
The probability that exactly 3 items in a sample of 200 are defective is approximately 0.061158.
The probability that exactly 3 items in a sample of 200 are defective is approximately 0.061158.

Q. 1 The Probabilify of getting an item deflective
is 0.005 . What is the probabicity trat exactly 3 itmy
in a sample of 200 are deffective? (Given:

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