Question

If the probability that a person suffers a bad reaction from an injection is $0.001$, then the probability that out of $2000$ individuals, exactly $3$ will suffer a bad reaction is:

• A. $\frac{4}{3} e^{-2}$
• B. $4 e^{-2}$
• C. $\frac{1}{3} e^{-2}$
• D. $2 e^{-2}$

Hint:

For very large $n$ and small $p$, the Poisson approximation $\lambda = np$ is the fastest and most efficient way to compute probabilities.

Answer 1

The Correct Option is A

Step 1: Concept
Since $n$ is very large and $p$ is very small, we use the Poisson distribution as an approximation to the Binomial distribution. The Poisson probability mass function is $P(X = r) = \frac{e^{-\lambda} \lambda^r}{r!}$, where $\lambda = np$.

Step 2: Meaning
Here, $n = 2000$ and $p = 0.001$. First, we compute the mean parameter $\lambda$.

Step 3: Analysis
Calculate $\lambda$: \[ \lambda = np = 2000 \times 0.001 = 2 \] Now, calculate the probability for exactly 3 individuals suffering a bad reaction ($r = 3$): \[ P(X = 3) = \frac{e^{-2} \cdot 2^3}{3!} = \frac{8 e^{-2}}{6} = \frac{4}{3} e^{-2} \]

Step 4: Conclusion
The probability that exactly 3 individuals suffer a bad reaction is $\frac{4}{3} e^{-2}$. Final Answer: (A)