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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.

Note: For simplicity, we will refer to the event of "coming down with a fever" simply as "fever" and the event of "feeling fatigued" simply as "fatigue". You may do the same.

Please answer the following three questions:

  1. Are the events of "fever" and "fatigue" complementary?
  2. Are the events of "fever" and "fatigue" mutually exclusive (or disjoint)?
  3. Are the events of "fever" and "fatigue" independent?

My answers:

  1. P(Fever) = 0.9 * 0.8 P(Fatigue) = 0.9 * 0.5 (I am thinking the answer is no, but don't know why)
  2. I think this answer is no because we are told there is a 90% chance that "at least one" of the two symptoms are shown, so there could be both. But I don't know how to show this using math.
  3. Yes they are independent

I am not sure how to approach this problem because the two probabilities given add up to more than 100%. Would I have to multiply 0.9 * 0.8 to figure out the probability of just getting a fever, and likewise for feeling fatigued? I need help showing the math behind these answers if I am even right.

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It's important to know what the different terms complement, mutually exclusive and independent mean mathematically.

  1. Two events $A, B$ are complementary if $P(A) +P(B)=1$.

  2. Two events are mutually exclusive if they cannot both happen together. In other words, $P(A\cap B)=0$

  3. Two events are independent if they don't affect each other. This means that $P(A\cap B) =P(A)P(B)$

Lets denote event $A$ as fever and $B$ as fatigue.

The question gives you: $$P(A\cup B) =0.9$$$$ P(A) =0.8$$$$P(B) =0.5$$

Clearly they are not complementary as $P(A)+P(B)=1.3>1$. Can you find $P(A\cap B)$ to determine if they are mutually exclusive, independent, or neither?

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