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A factory needs two raw materials.
The probability of not having an adequate supply of material A is 0.05, whereas
the probability of not having an adequate supply of material B is 0.03.
A study determines that the probability of a shortage in both A and B is 0.01.
(a) Let E be the event "shortage of A" and F be the event "shortage of B".
Construct a Venn diagram representing events E and F.
(b) Are events E and F independent explain
(c) What proportion of the time can the factory operate? Explain
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We are given P(E) = P(shortage of A) = 0.05;
P(F) = P(shortage of B) = 0.03;
P(E and F) = P((shortage of A) AND (shortage of B)) = 0.01.
It implies P(E)*P(F) = 0.05*0.03 = 0.0015. Compare it with P(E and F) = 0.01.
You see that P(E)*P(F) =/= P(E and F). Hence, the events E and F are NOT independent.
It is the ANSWER to question (b).
Next, P((shortage of A) OR (shortage of B)) = 0.05 + 0.03 - 0.01 = 0.07.
It implies P(no ((shortage of A) OR (shortage of B))) = 1 - 0.07 = 0.93. (*) (complementary event).
According to the context, the condition that the factory operates normally is
"no ((shortage of A) OR (shortage of B))".
The probability of it is 0.93, according to (*).
So, the factory will operate 93% of time.
It is the ANSWER to question (c).
Solved: questions (b) and (c) are answered.