S is the sample space and A, B are two events of a random experiment. Match the items of List A with the items of List B.
Then the correct match is:
Understanding the Matching of Events
Let's analyze each statement from List A and correctly match it with its corresponding expression from List B based on fundamental probability rules.
I. A, B are mutually exclusive events:
Mutually exclusive events cannot occur together, i.e.,
\[
A \cap B = \emptyset
\]
Therefore, the probability of their union is:
\[
P(A \cup B) = P(A) + P(B)
\]
Correct Match: (d) from List B.
II. A, B are independent events:
Independence implies that the occurrence of one does not affect the other. So:
\[
P(A \cap B) = P(A) \cdot P(B)
\]
Correct Match: (c) from List B.
III. \( A \cap B = A \):
This means that event A is completely contained within B (A ⊆ B). Then:
\[
P(A \cup B) = P(B)
\]
Correct Match: (b) from List B.
IV. \( A \cup B = S \):
The union of A and B covers the entire sample space, so:
\[
P(A \cup B) = 1
\]
Correct Match: (a) from List B.
Final Matching:
I → (d), II → (c), III → (b), IV → (a)
Correct Answer:
\[
\boxed{I - d, \quad II - c, \quad III - b, \quad IV - a}
\]
Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very healthy, 200 maintained average health and 100 had a poor health record.
Let \( A_1 \): People with good health,
\( A_2 \): People with average health,
and \( A_3 \): People with poor health.
During a pandemic, the data expressed that the chances of people contracting the disease from category \( A_1, A_2 \) and \( A_3 \) are 25%, 35% and 50%, respectively.
Based upon the above information, answer the following questions:
(i) A person was tested randomly. What is the probability that he/she has contracted the disease?}
(ii) Given that the person has not contracted the disease, what is the probability that the person is from category \( A_2 \)?