Probability distributions — binomial, Poisson, normal comparison

Question

When do we use binomial distribution vs Poisson vs normal distribution? Compare their properties and show how to pick the right one for a given problem.

(CBSE 12 + JEE Main — conceptual + numerical)

Solution — Step by Step

Feature	Binomial	Poisson	Normal
Type	Discrete	Discrete	Continuous
Parameters	$n$ (trials), $p$ (probability)	$\lambda$ (mean rate)	$\mu$ (mean), $\sigma$ (std dev)
Use when	Fixed trials, two outcomes	Rare events, large $n$ , small $p$	Large samples, symmetric data
Formula	$P(X=r) = \binom{n}{r}p^r q^{n-r}$	$P(X=r) = \frac{e^{-\lambda}\lambda^r}{r!}$	$f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
Mean	$np$	$\lambda$	$\mu$
Variance	$npq$	$\lambda$	$\sigma^2$

Use Binomial when:

There are a fixed number of trials $n$
Each trial has exactly two outcomes (success/failure)
Probability $p$ is constant across trials
Trials are independent

Example: A coin is tossed 10 times. Find $P(\text{exactly 3 heads})$ .

$P(X = 3) = \binom{10}{3}(0.5)^3(0.5)^7 = 120 \times \frac{1}{1024} = \mathbf{0.1172}$

Use Poisson when:

Events occur randomly in a given interval (time, area, volume)
The average rate $\lambda$ is known
Events are independent and $n$ is large, $p$ is small

Example: A call centre receives 4 calls per hour on average. Find $P(\text{exactly 6 calls in an hour})$ .

$P(X = 6) = \frac{e^{-4} \cdot 4^6}{6!} = \frac{0.01832 \times 4096}{720} = \mathbf{0.1042}$

Use Normal when:

Data is continuous and symmetric about the mean
You are dealing with large samples (Central Limit Theorem applies)
The problem gives $\mu$ and $\sigma$ (or asks for areas under the curve)

Convert to standard normal: $Z = \frac{X - \mu}{\sigma}$ , then use the Z-table.

flowchart TD
    A["Given a probability problem"] --> B{"Is the variable discrete or continuous?"}
    B -- Continuous --> C["Use Normal Distribution"]
    B -- Discrete --> D{"Fixed number of trials n?"}
    D -- Yes --> E{"Is p constant and trials independent?"}
    E -- Yes --> F["Use Binomial Distribution"]
    D -- No --> G{"Events per interval with known rate λ?"}
    G -- Yes --> H["Use Poisson Distribution"]
    F --> I{"Is n large and p small?"}
    I -- Yes --> J["Poisson can approximate Binomial"]
    F --> K{"Is np ≥ 5 and nq ≥ 5?"}
    K -- Yes --> L["Normal can approximate Binomial"]

Why This Works

These three distributions model different physical situations. Binomial counts successes in a fixed number of independent trials. Poisson counts random events in a continuous interval. Normal describes the bell-curve shape that arises whenever many small independent effects add up (Central Limit Theorem).

The approximations connect them: when $n$ is large and $p$ is small, binomial approaches Poisson (with $\lambda = np$ ). When $n$ is large enough that $np \geq 5$ and $nq \geq 5$ , binomial approaches normal (with $\mu = np$ , $\sigma = \sqrt{npq}$ ).

Alternative Method

For JEE Main, most problems are Binomial. The quick check: if the question says “n trials” or “n times” with a success probability, it is Binomial. If it says “on average, X events per unit time,” it is Poisson. If it gives mean and standard deviation of a large population, it is Normal.

Common Mistake

Students mix up the conditions for Poisson approximation to Binomial. It requires $n \geq 20$ AND $p \leq 0.05$ (roughly). If $n = 10$ and $p = 0.3$ , using Poisson gives a poor approximation. Stick with exact Binomial when $n$ is small or $p$ is not near zero.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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