Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 2 / 9F (Probability)

Table of contents
- i 9F
  - Solution i
- ii 9F
  - Solution ii
- iii 9F
  - Solution iii
- iv 9F
  - Solution iv

i

Solution

Each of the

(3 n)

triples is a triangle with probability

p^{3}

, so

E T = (3 n) p^{3}

.

ii

Solution

Markov inequality gives

P (T > 0) \leq E T = O (n^{3 - 3 α}) \to 0

when

α > 1

.

iii

Solution

Chebyshev inequality gives

P (T = 0) \leq P (∣ T - E T ∣ \geq E T) \leq var (T) / (E T)^{2} \to 0

.

iv

Solution

Only triangle pairs sharing an edge have nonzero covariance, so

var T = (3 n) (p^{3} - p^{6}) + 2 (2 n) (2 n - 2) (p^{5} - p^{6}) .

After division by

(E T)^{2}

this is

O (n^{- 3} p^{- 3} + n^{- 2} p^{- 1}) \to 0

for

p = n^{- α}

,

0 < α < 1

; use part (iii).

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