Question

Villages A, B, C and D are connected by overhead telephone lines joining AB,
AC, BC, BD and CD. As a result of severe gales, there is a probability p (the
same for each link) that any particular link is broken.
(a) Show that the probability that a call can be made from A to B is
1 − p2 − 2p3 + 3p4 − p5
.
(b) Show that the probability that a call can be made from D to A is
1 − 2p2 − 2p3 + 5p4 − 2p5
.

Answer 1

Answer:

(a) Probability of a call failure from A to B $=p^{2}+2 p^{3}-3 p^{4}+p^{5}$

(b) Probability of call failure from D to A $=p^{2}+4 p^{3}-6 p^{4}+2 p^{5}$

Step-by-step explanation:

(a) Probability of a call failure from A to B

$=[\mathrm{AB}(\mathrm{x}), \mathrm{AC}(\mathrm{x})]+[\mathrm{AB}(\mathrm{x}), \mathrm{AC}(\mathrm{ok}), \mathrm{CB}(\mathrm{x}), \mathrm{CD}(\mathrm{ok}), \mathrm{DB}(\mathrm{x})]+[\mathrm{AB}(\mathrm{x}), \mathrm{AC}(\mathrm{ok}), \mathrm{CB}(\mathrm{x}), \mathrm{CD}(\mathrm{x})]$

$=p^{2}+p^{3}(1-p)^{2}+p^{3}(1-p)$

$=p^{2}+2 p^{3}-3 p^{4}+p^{5}$

So, the probability of success $=p^{2}+2 p^{3}-3 p^{4}+p^{5}$.

(b) Probability of call failure from D to A

$=[\mathrm{DB}(\mathrm{x}), \mathrm{DC}(\mathrm{x})]+[\mathrm{DB}(\mathrm{x}), \mathrm{DC}(\mathrm{ok}), \mathrm{CA}(\mathrm{x}), \mathrm{CB}(\mathrm{x})]+[\mathrm{DB}(\mathrm{x}), \mathrm{DC}(\mathrm{ok}), \mathrm{CA}(\mathrm{x}), \mathrm{CB}(\mathrm{ok}), \mathrm{BA}(\mathrm{x})]$$+[\mathrm{DC}(\mathrm{x}), \mathrm{DB}(\mathrm{ok}), \mathrm{BA}(\mathrm{x}), \mathrm{BC}(\mathrm{x})]+[\mathrm{DC}(\mathrm{x}), \mathrm{DB}(\mathrm{ok}), \mathrm{BA}(\mathrm{x}), \mathrm{BC}(\mathrm{ok}), \mathrm{CA}(\mathrm{x})]$

$=p^{2}+p^{3}(1-p)+p^{3}(1-p)^{2}+p^{3}(1-p)+p^{3}(1-p)^{2}$

$=p^{2}+4 p^{3}-6 p^{4}+2 p^{5}$.

Therefore,

Probability of a call failure from A to B $=p^{2}+2 p^{3}-3 p^{4}+p^{5}$.

Probability of call failure from D to A $=p^{2}+4 p^{3}-6 p^{4}+2 p^{5}$.