All subjects · Mathematics · WST01 (S1)

Normal probabilities in context

Real questions give X ~ N(μ, σ2) in context. Standardise, use the table and symmetry, then answer in the language of the question.

Method

1. Write the probability in terms of X, such as P(X > 45).

2. Standardise: P(Z > (45 - 50)/4) = P(Z > -1.25).

3. Sketch and use symmetry: P(Z > -1.25) = Φ(1.25) = 0.8944.

Show the standardisation clearly: examiners award a method mark for it.

Between two values

P(44 < X < 58) = P(-1.5 < Z < 2) = Φ(2) - (1 - Φ(1.5)) = 0.9772 - 0.0668 = 0.9104.

Expected numbers and conditional probability

Expected number = n × probability. Out of 200 items, the number expected above 56 is 200 × 0.0668 = 13.4, so about 13.

Conditional: P(X > 56 | X > 50) = P(X > 56)/P(X > 50) = 0.0668/0.5 = 0.1336, because X > 56 is inside X > 50.

Give final answers to 3 s.f. unless told otherwise; table probabilities are 4 d.p.

Worked example

The mass of a bag of flour, X grams, has X ~ N(1010, 52). Find the probability that a bag weighs less than 1000 g.

  1. P(X < 1000) = P(Z < (1000 - 1010)/5) = P(Z < -2)
  2. = 1 - Φ(2) = 1 - 0.9772

Answer: 0.0228

Key idea: Standardise, sketch, use the table and symmetry; expected number = n × p; P(A | B) = P(A)/P(B) when A is inside B.

Check yourself

  1. X ~ N(50, 42). Find P(X < 56). Give your answer to 4 decimal places.
    Show answer

    0.9332. z = (56 - 50)/4 = 1.5; Φ(1.5) = 0.9332.

  2. X ~ N(50, 42). Find P(X > 45). Give your answer to 4 decimal places.
    Show answer

    0.8944. z = -1.25; P(Z > -1.25) = Φ(1.25) = 0.8944.

  3. X ~ N(50, 42). Find P(44 < X < 58). Give your answer to 4 decimal places.
    Show answer

    0.9104. z values -1.5 and 2: 0.9772 - 0.0668 = 0.9104.

  4. X ~ N(1010, 52). Find P(X < 1000). Give your answer to 4 decimal places.
    Show answer

    0.0228. z = -2; 1 - Φ(2) = 0.0228.

  5. X ~ N(50, 42). Out of 200 items, how many would you expect to have X > 56? Give your answer to 3 significant figures.
    Show answer

    13.4. P(X > 56) = 1 - Φ(1.5) = 0.0668; 200 × 0.0668 = 13.36.

  6. X ~ N(50, 42). Find P(X > 56 | X > 50). Give your answer to 4 decimal places.
    Show answer

    0.1336. P(X > 56)/P(X > 50) = 0.0668/0.5 = 0.1336.

  7. Heights of adults, H cm, are modelled by H ~ N(170, 82). Find P(H > 180). Give your answer to 4 decimal places.
    Show answer

    0.1056. z = 10/8 = 1.25; 1 - 0.8944 = 0.1056.

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Last updated 7 November 2026. Spotted a mistake? Tell us.