Question

Two symptoms are associated with a certain disease.

There is a 95% probability that at least one of the symptoms occurs;
in addition, the first symptom occurs with 50% probability, the second symptom occurs with 45% probability.

Based on these probability results, answer the following two questions

1) Are the two events "first symptom occurs" and "second symptom occurs" mutually exclusive (i.e. disjoint)?

2) Are the two events "first symptom occurs" and "second symptom occurs" independent?

For each question, clearly state YES or NO and provide a brief written explanation that includes the appropriate numerical support.

Answer 1

Mutually exclusive, dependent events

Explanation:

Two events A and B are mutually exclusive if
`P(A\cap B)=0`

Two events A and B are independent if
`P(A\cap B)=P(A)\cdot P(B)`

Remark: All mutually exclusive events are dependent.

Now,

A = the first symptom occurs

B = the second symptom occurs

`P(A)=0.5\ (\text{or } 50\%)`

`P(B)=0.45\ (\text{or } 45\%)`

`P(A\cup B)=0.95 \ (\text{or }95\%)`

Use the rule

`P(A\cup B)=P(A)+P(B)-P(A\cap B)\\ \\0.95=0.5+0.45-P(A\cap B)\\ \\P(A\cap B)=0.5+0.45-0.95=0.95-0.95=0`

Thus, the events A and B are mutually exclusive (disjoint) and dependent (accordint to the remark)