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Comparing normal variables: P(X > Y) and capacity problems

Edexcel IAL Mathematics · WST03 Statistics 3

Many S3 questions ask whether one random quantity exceeds another, or whether a total exceeds a limit. Turn the comparison into a single combined variable and compare it with 0 or the limit.

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P(X > Y)

Rewrite as P(X - Y > 0). Then X - Y ~ N(μx - μy, σx2 + σy2).

Example: task times A ~ N(25, 32) and B ~ N(22, 42), independent. A - B ~ N(3, 25), so P(A > B) = P(Z > -0.6) = 0.7257.

Multiples

P(X > 2Y) becomes P(X - 2Y > 0). With X ~ N(50, 62) and Y ~ N(20, 22): X - 2Y ~ N(10, 36 + 4 × 4) = N(10, 52). Notice it is 22 × Var(Y), because 2Y is one value of Y doubled.

For 'within' questions use the modulus: P(|X - Y| < k) = P(-k < X - Y < k).

Capacity and fit problems

Lift with 10 passengers each W ~ N(75, 122) kg and a limit of 800 kg: the total is a sum of 10 separate people, ~ N(750, 1440).

A rod fits a hole if hole diameter minus rod diameter is positive: use H - R and find P(H - R > 0). Always state the combined distribution, with its mean and variance, before standardising: the mark scheme gives a mark for the fully correct distribution.

Worked example

X ~ N(50, 62) and Y ~ N(20, 22) are independent. Find P(X > 2Y).

  1. P(X > 2Y) = P(X - 2Y > 0)
  2. X - 2Y ~ N(50 - 40, 36 + 16) = N(10, 52)
  3. P(X - 2Y > 0) = P(Z > -10/sqrt(52)) = P(Z < 1.3868)

Answer: P = 0.9177 (tables)

Key idea: Move everything to one side: P(X > Y) = P(X - Y > 0). Then combine means and add variances (squared constants).

Check yourself

  1. Task times A ~ N(25, 32) and B ~ N(22, 42) minutes are independent. Find P(A > B). Give your answer to 4 decimal places.
    Show answer

    0.7257. A - B ~ N(3, 25); P(A - B > 0) = P(Z > -0.6) = 0.7257.

  2. X ~ N(50, 62) and Y ~ N(20, 22) are independent. Find Var(X - 2Y).
    Show answer

    52. 36 + 22 × 4 = 52.

  3. X ~ N(50, 62) and Y ~ N(20, 22) are independent. Find P(X > 2Y). Give your answer to 4 decimal places.
    Show answer

    0.9172. X - 2Y ~ N(10, 52); P = P(Z < 1.3868) = 0.9177.

  4. Adults have mass W ~ N(75, 122) kg. A lift has a limit of 800 kg. Find the probability that 10 randomly chosen adults exceed the limit. Give your answer to 4 decimal places.
    Show answer

    0.0938. Total ~ N(750, 1440); z = 50/sqrt(1440) = 1.3176; P = 0.0934.

  5. Two pieces of string have lengths L1 and L2, independent, each ~ N(40, 0.52) cm. Find the probability that their lengths differ by less than 1 cm. Give your answer to 4 decimal places.
    Show answer

    0.8427. L1 - L2 ~ N(0, 0.5); P(-1 < L1 - L2 < 1) = 2Φ(1.4142) - 1 = 0.8414.

  6. X ~ N(50, 62) and Y ~ N(20, 22) are independent. What is the distribution of X - 2Y?
    • N(10, 52)
    • N(10, 44)
    • N(10, 28)
    • N(10, 20)
    Show answer

    N(10, 52). Mean 50 - 40 = 10; variance 36 + 4 × 4 = 52.

  7. Rod diameters R ~ N(2.00, 0.022) cm and hole diameters H ~ N(2.05, 0.032) cm. Which probability gives the chance a random rod fits a random hole?
    • P(R > 2.05)
    • P(H > 2.00)
    • P(H - R > 0)
    • P(H - R > 0.05)
    Show answer

    P(H - R > 0). The rod fits when the hole is wider than the rod, so H - R > 0.

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Last updated 8 November 2026. Spotted a mistake? Tell us.