Probability and Distributions Questions for SSC CGL

Basic Probability Concepts

Probability measures the likelihood of an event on a scale from 0 to 1. The sample space S is the set of all possible outcomes. An event A is any subset of S. Classical probability assumes all outcomes are equally likely. Mutually exclusive events cannot happen together. Independent events do not affect each other's probability.

• P(A) = Number of favourable outcomes / Total outcomes (classical definition)
• 0 <= P(A) <= 1 always
• P(A) + P(A complement) = 1
• For mutually exclusive events: P(A or B) = P(A) + P(B)
• For independent events: P(A and B) = P(A) x P(B)
• Addition rule general: P(A or B) = P(A) + P(B) - P(A and B)

Conditional Probability and Bayes Theorem

Conditional probability P(A|B) means the probability of A given that B has already occurred. It restricts the sample space to B. Bayes theorem flips the conditioning — it finds P(cause | effect) when you know P(effect | cause). SSC CGL tests straightforward Bayes theorem problems with 2 to 3 hypotheses.

• P(A|B) = P(A and B) / P(B), provided P(B) > 0
• For independent events: P(A|B) = P(A) — knowledge of B changes nothing
• Bayes theorem: P(Hi|A) = P(Hi) x P(A|Hi) / sum of [P(Hj) x P(A|Hj)]
• Total Probability: P(A) = sum of P(A|Hi) x P(Hi) over all partitions Hi
• Posterior probability = updated belief after seeing evidence

Binomial Distribution

Use Binomial when you have n independent trials, each with exactly two outcomes (success/failure), and constant probability p of success. The classic examples are coin tosses, defective items in a batch, and pass/fail scenarios. SSC CGL frequently asks for mean, variance, and P(X = r).

• Notation: X ~ B(n, p), where n = trials, p = success probability, q = 1 - p
• P(X = r) = nCr x p^r x q^(n-r), for r = 0, 1, 2, ..., n
• Mean = np
• Variance = npq
• Standard Deviation = sqrt(npq)
• Binomial is symmetric when p = 0.5, positively skewed when p < 0.5

Poisson Distribution

Poisson distribution applies when events occur rarely but over a large number of trials — like number of accidents per day, defects per unit, or calls per hour. It is actually a limiting case of Binomial when n is very large and p is very small, such that np = lambda remains constant. SSC CGL tests mean, variance, and simple probability calculations.

• Parameter: lambda (λ) = average rate of occurrence
• P(X = r) = e^(-λ) x λ^r / r!, for r = 0, 1, 2, ...
• Mean = λ, Variance = λ (both are equal — key exam fact)
• Standard Deviation = sqrt(λ)
• Poisson approximates Binomial when n > 50 and p < 0.1, use λ = np
• Poisson distribution is always positively skewed

Normal Distribution

The Normal distribution is the most important continuous distribution. It is bell-shaped, symmetric about its mean, and fully defined by mean (μ) and standard deviation (σ). SSC CGL tests properties, the empirical rule (68-95-99.7), and standard normal Z-score conversion. Area under the curve represents probability.

• Notation: X ~ N(μ, σ²)
• Mean = Median = Mode (perfectly symmetric)
• Total area under curve = 1
• Empirical Rule: 68% within μ ± 1σ, 95% within μ ± 2σ, 99.7% within μ ± 3σ
• Standard Normal Z = (X - μ) / σ, has mean 0 and variance 1
• Normal curve is asymptotic to x-axis — never touches it

• Confusing Variance and Standard Deviation — exam often gives one and asks the other. Remember Variance = σ², SD = σ. For Binomial, Variance = npq not sqrt(npq).
• Using Binomial formula when Poisson is needed — if n is large and p is tiny (like defect rate 0.01 with n=500), always switch to Poisson with λ = np.
• Forgetting that in Poisson distribution Mean = Variance = λ. If a question gives Mean = 4, Variance is automatically 4 — do not recalculate.
• In Normal distribution, mixing up area in both tails vs one tail. P(|Z| < 1) = 0.6827, but P(Z < 1) = 0.8413. SSC questions specify direction — read carefully.
• Applying the addition rule P(A+B) = P(A)+P(B) without subtracting intersection for non-mutually-exclusive events — a very common slip under time pressure.

• BPN order of distributions: B for Binomial (two outcomes), P for Poisson (rare events, mean = variance), N for Normal (bell curve, symmetric). Remember BPN as Buy Packets Now.
• Binomial variance trick: V = npq. Think N-P-Q: N trials, P success, Q failure. Variance needs all three.
• For Normal distribution 68-95-99.7 Rule: Remember it as 1-sigma = 68 (one year), 2-sigma = 95 (two years), 3-sigma = 99.7 (three years of study pays off).
• Bayes theorem memory: Prior x Likelihood divided by Evidence. PLdE = Prior times Likelihood divided by Evidence total.

• SSC CGL Tier 2 Statistics paper typically has 3 to 4 questions directly from distributions. Binomial and Normal appear more frequently than Poisson. Prioritize Binomial mean/variance and Normal Z-score conversion.
• Property-based questions are very common — Is Mean = Median = Mode? Which distribution has equal Mean and Variance? These are 30-second questions if you memorize the properties.
• Bayes theorem questions in SSC usually involve 2 boxes or 2 factories — set up a small table with Prior, Likelihood, and Joint probability columns to avoid errors under pressure.
• For Binomial, if the question says coin tossed n times or item selected with replacement, it is always Binomial. Without replacement means Hypergeometric — but SSC rarely tests that at this level.
• In Normal distribution questions, if Z-table values are given in the question itself (like P(Z<1.5) = 0.9332), use them directly — do not waste time recalling tables. SSC often provides necessary values.