Theoretical DistributionsPYQ Sept 25Question 4186 of 230
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An emergency room receives an average of 3 patients per hour. What is the probability that exactly 2 patients arrive in an hour? (Given: e0=1\displaystyle e^0 = 1, e−1=0.367\displaystyle e^{-1} = 0.367, e−2=0.135\displaystyle e^{-2} = 0.135, e−3=0.049\displaystyle e^{-3} = 0.049, e−4=0.018\displaystyle e^{-4} = 0.018, e−5=0.0067\displaystyle e^{-5} = 0.0067)

Options

A0.22
B0.3
C0.27
D0.25
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Correct Answer

✅ Option a — 0.22

All Options:

  • A0.22
  • B0.3
  • C0.27
  • D0.25

Detailed Solution & Explanation

The arrival of patients in an emergency room follows a Poisson distribution. The probability mass function of a Poisson distribution is:
P(X=x)=e−λλxx!P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}
where λ\displaystyle \lambda is the mean rate of arrivals per unit time, and x\displaystyle x is the number of arrivals.
1. **Given Data**:
- λ=3\displaystyle \lambda = 3 patients per hour
- x=2\displaystyle x = 2 patients
- e−3=0.049\displaystyle e^{-3} = 0.049
2. **Calculation**:
Substitute the values into the Poisson formula:
P(X=2)=e−3×322!P(X = 2) = \frac{e^{-3} \times 3^2}{2!}
P(X=2)=0.049×92P(X = 2) = \frac{0.049 \times 9}{2}
P(X=2)=0.4412=0.2205P(X = 2) = \frac{0.441}{2} = 0.2205
Rounding to two decimal places, we get 0.22\displaystyle 0.22.
Hence, **Option A** is the correct answer.

About This Chapter: Theoretical Distributions

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Binomial, Poisson, Normal Distribution

This chapter covers Binomial, Poisson, Normal Distribution and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

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