1992 Paper 1 Q16

Year: 1992
Paper: 1
Question Number: 16

Course: LFM Stats And Pure
Section: Independent Events

Difficulty: 1500.0 Banger: 1504.2
probability independent events graph theory network reliability road network complementary counting optimisation

Problem

The four towns \(A,B,C\) and \(D\) are linked by roads \(AB,AC,CB,BD\) and \(CD.\) The probability that any one road will be blocked by snow on the 1st of January is \(p\), independent of what happens to any other \([0 < p < 1]\). Show that the probability that any open route from \(A\) to \(D\) is \(ABCD\) is \[ p^{2}(1-p)^{3}. \] In order to increase the probability that it is possible to get from \(A\) to \(D\) by a sequence of unblocked roads the government proposes either to snow-proof the road \(AB\) (so that it can never be blocked) or to snow-proof the road \(CB.\) Because of the high cost it cannot do both. Which road should it choose (or are both choices equally advantageous)? In fact, \(p=\frac{1}{10}\) and the government decides that it is only worth going ahead if the present probability of \(A\) being cut off from \(D\) is greater than \(\frac{1}{100}.\) Will it go ahead?

No solution available for this problem.

Rating Information

Difficulty Rating: 1500.0

Difficulty Comparisons: 0

Banger Rating: 1504.2

Banger Comparisons: 4

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Problem source
The four towns $A,B,C$ and $D$ are linked by roads $AB,AC,CB,BD$
and $CD.$ The probability that any one road will be blocked by snow
on the 1st of January is $p$, independent of what happens to any
other $[0 < p < 1]$. Show that the probability that any open route from
$A$ to $D$ is $ABCD$ is 
\[
p^{2}(1-p)^{3}.
\]
In order to increase the probability that it is possible to get from
$A$ to $D$ by a sequence of unblocked roads the government proposes
either to snow-proof the road $AB$ (so that it can never be blocked)
or to snow-proof the road $CB.$ Because of the high cost it cannot
do both. Which road should it choose (or are both choices equally
advantageous)? 

In fact, $p=\frac{1}{10}$ and the government decides that it is only
worth going ahead if the present probability of $A$ being cut off
from $D$ is greater than $\frac{1}{100}.$ Will it go ahead?
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