Question 1186927
Let's analyze the given events related to mother's age at birth:

**a. Are events A, B, C, and D pairwise mutually exclusive?**

**Yes.**

*Explanation:* Pairwise mutually exclusive means that no two events can occur at the same time.  A mother cannot be both under 20 *and* between 20-24 years old at the same time. The age ranges are distinct and non-overlapping. Therefore, the events A, B, C, and D are pairwise mutually exclusive.

**b. State in words the event E = (A ∪ B)**

Event E represents the event that the mother is *under 25 years old* at the time of giving birth.  (It's the union of the events "under 20" and "20-24").

**c. State in words the event F = (B ∪ C)**

Event F represents the event that the mother is *between 20 and 29 years old* (inclusive) at the time of giving birth. (It's the union of the events "20-24" and "25-29").

**d. Comment on the event G = (A ∩ B)**

Event G represents the intersection of events A and B.  Since A and B are mutually exclusive (as discussed in part a), their intersection is the empty set.  That is, it is impossible for a mother to be both under 20 and between 20 and 24 at the same time. So, G is the *null event* or *impossible event*.  P(G) = 0.

**e. Comment on the event H = (A ∪ (Dᶜ))**

*   Dᶜ (D complement or "D bar") represents the event that the mother is *not* between 30 and 44 years old. This includes mothers under 20, between 20-24 and 25-29.  (It's the union of A, B, and C)

*   H = (A ∪ (Dᶜ)) represents the union of event A (under 20) and the event (Dᶜ) (not between 30 and 44). Since A is already included within Dᶜ,  H is equivalent to Dᶜ.

Therefore, event H represents the event that the mother is *not* between 30 and 44 years old at the time of giving birth (i.e., the mother is 29 or younger).