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Class 12 Statistics Notes · GSEB

Finding Probability

Finding Probability — solve P(a < X < b) in three steps: convert to Z, read the table, subtract. GSEB Class 12 Commerce Statistics notes with guided solver.

Last updated: 22 Sep 2026

Notes

The Three-Step Method

Every normal probability question — whatever the story about marks, heights or delivery times — solves in the same three moves: convert to Z → read the table → subtract.

Given

1/4

You are told X ~ N(50, 10²) and asked for P(45 < X < 60).

Two parameters (μ, σ) + a range — that is everything the question gives you.

The Formula Toolkit

Between Two Values

P(a<X<b)=Φ(Zb)−Φ(Za)P(a < X < b) = \Phi(Z_b) - \Phi(Z_a)

Upper Tail

P(X>a)=1−Φ(Za)P(X > a) = 1 - \Phi(Z_a)

Lower Tail

P(X<a)=Φ(Za)P(X < a) = \Phi(Z_a)

Symmetry Rule

Φ(−Z)=1−Φ(Z)\Phi(-Z) = 1 - \Phi(Z)

Reading the notation

Φ(Z) means "the area to the LEFT of Z under the standard normal curve" — exactly what the Z-table prints. So P(a < X < b) = (area left of Zb) − (area left of Za) = the strip in between.

Try It: Step-by-Step Solver

Set your own problem (or keep the default), then pass each checkpoint — the next step unlocks only when you get the current one right.

Guided Solver — P(40 < X < 70) for X ~ N(50, 10²)

Formula: Z = (X − μ) / σ. Compute both Z-values and type them (2 decimal places):

Solved Examples

Solved Example

Problem

Marks ~ N(50, 10²). Find P(X > 65).

Solution

P(X > 65) = 0.0668 — about 6.7% of students

Solved Example

Problem

Zomato delivery times in your city are N(32, 6²) minutes. What is the probability an order arrives within 40 minutes?

Solution

P = 0.9082 — a 90.8% chance the rider makes it in 40 minutes

Solved Example

Problem

Heights ~ N(165, 7²) cm. Find P(X < 158).

Solution

P = 0.1587 — about 15.9% of people are below 158 cm

Cheat Sheet: Which Formula When?

Match the question to the formula — this is the entire procedure in one table.
Question asks for…FormulaExample (Z = 1.50 case)
Between two valuesΦ(Zb) − Φ(Za)P = Φ(1.00) − Φ(−0.50)
Greater than a value1 − Φ(Za)P = 1 − Φ(1.50) = 0.0668
Less than a valueΦ(Za)P = Φ(1.50) = 0.9332
Negative Z anywhereΦ(−Z) = 1 − Φ(Z)Φ(−1.50) = 1 − 0.9332 = 0.0668
Beyond ±3.49 in the table≈ 0.9997 / 0.0003Φ(3.50) ≈ 0.99977

Key Takeaways

Key Takeaways

  • Always the same three steps: Z = (X − μ)/σ → look up Φ in the table → subtract in the right order.
  • Φ(Z) is the area to the LEFT of Z — so P(between) = bigger Φ − smaller Φ, P(above) = 1 − Φ, P(below) = Φ.
  • For negative Z use Φ(−Z) = 1 − Φ(Z) — never read a negative row off a standard table.
  • Round Z to 2 decimal places: row = first two digits, column = the hundredth — the highlighted row in the solver shows this exactly.
  • Sketch the curve and shade the range before calculating; order-of-subtraction errors vanish once you can see the slice.

Practice

  1. X ~ N(70, 5²). Find P(65 < X < 80).
  2. Wages of workers ~ N(₹18000, ₹1500²). Find P(X > ₹21000).
  3. Find P(X < 55) for N(50, 10²) using the symmetry rule — write each step.

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