1998 Paper 3 Q12

Year: 1998
Paper: 3
Question Number: 12

Course: LFM Stats And Pure
Section: Independent Events

Difficulty: 1700.0 Banger: 1482.8

probability independent events complementary counting graph theory tetrahedron connectivity inclusion-exclusion polynomial

Problem

The mountain villages $A,B,C$ and $D$ lie at the vertices of a tetrahedron, and each pair of villages is joined by a road. After a snowfall the probability that any road is blocked is $p$, and is independent of the conditions of any other road. The probability that, after a snowfall, it is possible to travel from any village to any other village by some route is $P$. Show that $$ P =1- p^2(6p^3-12p^2+3p+4). $$ %In the case $p={1\over 3}$ show that this probability is ${208 \over 243}$.

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

Difficulty Comparisons: 0

Banger Rating: 1482.8

Banger Comparisons: 3

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Problem source

The mountain villages $A,B,C$ and $D$ lie at the vertices of a
tetrahedron, and each pair of villages is joined by a road. After
a snowfall the probability that any road is blocked is $p$, and 
is independent of the conditions of any other road. The 
probability that, after a snowfall,
 it is possible to travel from any village to
any other village by some route is $P$.  Show that 
$$
P =1- p^2(6p^3-12p^2+3p+4).
$$
%In the case $p={1\over 3}$ show that this probability is ${208 \over 243}$.

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