If at least one of four products is defective, this is event…
If at least one of four products is defective, this is event $A$; if no fewer than two products are defective, this is event $B$. What do the complementary (opposite) events $\overline{A}$ and $\overline{B}$ mean?
Explanation of Method
Everything depends only on the number $k$ of defective products, which can be $0, 1, 2, 3$ or $4$. We write $A$ and $B$ as sets of values of $k$. The complementary event $\overline{X}$ consists of all values not in $X$, so we just take the remaining values of $k$ and translate them back into words. A useful rule: the opposite of "at least $m$" is "at most $m-1$" (i.e. "fewer than $m$").
Step 1: The possible outcomes
Let $k$ be the number of defective products among the four. Then
\[ k \in \{0, 1, 2, 3, 4\}. \]
The phrase "at least one" means one or more; "no fewer than two" means two or more.
Step 2: The events $A$ and $B$
\[ A = \{\text{at least one defective}\} = \{k \ge 1\} = \{1,2,3,4\}, \]
\[ B = \{\text{no fewer than two defective}\} = \{k \ge 2\} = \{2,3,4\}. \]
Step 3: What a complementary event is
The complementary (opposite) event $\overline{X}$ occurs exactly when $X$ does not occur. It contains all outcomes not in $X$, and
\[ X + \overline{X} = \Omega \ \ (\text{one of them always occurs}), \qquad X\overline{X} = \varnothing \ \ (\text{they never occur together}). \]
Step 4: The event $\overline{A}$
Remove $\{1,2,3,4\}$ from $\{0,1,2,3,4\}$; only $k=0$ is left:
\[ \overline{A} = \{k = 0\}. \]
\[ \boxed{\overline{A} = \text{"none of the four products is defective" = "all four products are good".}} \]
Check in words: the opposite of "at least one is defective" is "not even one is defective".
Step 5: The event $\overline{B}$
Remove $\{2,3,4\}$ from $\{0,1,2,3,4\}$; the values $k=0$ and $k=1$ are left:
\[ \overline{B} = \{k \le 1\} = \{0, 1\}. \]
\[ \boxed{\overline{B} = \text{"at most one product is defective" = "fewer than two are defective".}} \]
That is: either all four products are good, or exactly one is defective. Equivalently, at least three products are good.
Step 6: Summary table
| $k$ (defective) | $A$ | $B$ | $\overline{A}$ | $\overline{B}$ |
|---|---|---|---|---|
| $0$ | no | no | yes | yes |
| $1$ | yes | no | no | yes |
| $2$ | yes | yes | no | no |
| $3$ | yes | yes | no | no |
| $4$ | yes | yes | no | no |
In every row exactly one of $A$, $\overline{A}$ says "yes", and exactly one of $B$, $\overline{B}$ says "yes" — as it must be for opposite events.
Step 7: Relations between the events
- $B\subseteq A$: if at least two products are defective, then certainly at least one is.
- $\overline{A}\subseteq\overline{B}$: if no product is defective, then certainly at most one is. Taking complements always reverses an inclusion: $B\subseteq A \iff \overline{A}\subseteq\overline{B}$.
- $A\overline{B} = \{k=1\}$: "exactly one product is defective" — this is the part of $\overline{B}$ that is not in $\overline{A}$, so $\overline{B} = \overline{A} + A\overline{B}$.
Step 8: A common mistake
The opposite of "at least two are defective" is not "at most two are defective" and not "none is defective". The value $k=2$ belongs to $B$, so it cannot belong to $\overline{B}$; and $k=1$ is not in $B$, so it must be in $\overline{B}$. The correct opposite is "at most one" ($k = 0$ or $1$).
Result
| Event | Values of $k$ | Meaning |
|---|---|---|
| $A$ | $k\ge 1$ | at least one product is defective |
| $\overline{A}$ | $k = 0$ | none of the four products is defective (all are good) |
| $B$ | $k\ge 2$ | at least two products are defective |
| $\overline{B}$ | $k\le 1$ | at most one product is defective (0 or 1) |