Question

A certain illness has two symptoms associated with it - a fever and fatigue. There is a 90% probability that at least one of the two symptoms occurs for a randomly selected person with the illness. There is an 80% probability that a randomly selected person with the illness will come down with a fever and there is a 50% probability that a randomly selected person with the illness will feel fatigued.

Please answer the following three questions:
1) Are the events of "fever" and "fatigue" mutually exclusive (or disjoint)?
2) Are the events of "fever" and "fatigue" complementary?
3) Are the events of "fever" and "fatigue" independent?

Answer 1

P(fever) = 0.80

P(fatigue) = 0.50

P(fever or fatigue) = 0.90

P(A or B) = P(A) + P(B) - P(A & B)

So, 0.9 = 0.8 + 0.5 - P(fever and fatigue)

P(fever and fatigue) = 0.8 + 0.5 - 0.9

= 0.4

1) Two events are mutually exclusive if both of the events cannot take place simultaneously

P(fever and fatigue) $\small \neq$ 0

So, the events are not mutually exclusive

2) Two events are complementary if P(A & B) $\small \neq$ 0 and P(A) + P(B) = 1

The events are not complementary

3) A and B are independent if P(A) x P(B) = P(A&B)

Here, P(fever) x P(fatigue) = 0.8x0.5 = 0.4

P(fever and fatigue) = 0.4

So, fever and fatigue are independent