A manufacturing company produces medicine bottles. It finds that probability of producing 3 defective medicine bottles is 2​/3 times probability of producing 4 defective medicine bottles.(Assuming that the number of defective medicine bottles in the company follows the Poisson distribution.) Find average number of defective bottles

To solve this problem, we use the fact that the number of defective medicine bottles follows a Poisson distribution. The probability of observing \( k \) defective bottles is given by:

\[
P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}
\]

where:
- \( \lambda \) is the average number of defective bottles (the parameter we need to find).
- \( k \) is the number of defective bottles.

Given:
\[
P(X = 3) = \frac{2}{3} P(X = 4)
\]

### Step 1: Express the probabilities using the Poisson distribution formula

\[
P(X = 3) = \frac{e^{-\lambda} \lambda^3}{3!}
\]
\[
P(X = 4) = \frac{e^{-\lambda} \lambda^4}{4!}
\]

### Step 2: Set up the equation using the given ratio

We are told that:

\[
\frac{e^{-\lambda} \lambda^3}{3!} = \frac{2}{3} \times \frac{e^{-\lambda} \lambda^4}{4!}
\]

### Step 3: Simplify the equation

Since the \( e^{-\lambda} \) terms cancel out, and substituting \( 3! = 6 \) and \( 4! = 24 \):

\[
\frac{\lambda^3}{6} = \frac{2}{3} \times \frac{\lambda^4}{24}
\]

Simplify further:

\[
\frac{\lambda^3}{6} = \frac{2 \lambda^4}{72} 
\]

\[
\frac{\lambda^3}{6} = \frac{\lambda^4}{36} 
\]

Now, multiply both sides by 36 to get rid of the denominator:

\[
6 \lambda^3 = \lambda^4
\]

### Step 4: Solve for \( \lambda \)

Divide both sides by \( \lambda^3 \) (assuming \( \lambda \neq 0 \)):

\[
6 = \lambda
\]

So, the average number of defective bottles \( \lambda \) is **6**.

Would you like more details or have any questions?  

Here are some related questions:
1. What would be the probability of producing exactly 5 defective bottles?
2. How does the average number of defective bottles affect overall production quality?
3. What is the probability of producing at most 4 defective bottles?
4. How would the solution change if the ratio was different?
5. What real-world factors might cause the Poisson distribution to be an inaccurate model in this context?

**Tip:** Always check the assumptions of your distribution model to ensure it applies to the situation at hand.

Explore how to find the average number of defective medicine bottles using the Poisson distribution. This detailed solution involves calculating probabilities for specific defective counts and determining the average λ. Suitable for advanced high school and undergraduate students.

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