Let's break down the solution step-by-step to find the probability that both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occur, given that [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are exhaustive events.
1. Understanding Exhaustive Events:
- Exhaustive events are defined as a set of events in a sample space such that at least one of these events must occur. The occurrence of these events covers the entire sample space.
- Mathematically, if [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are exhaustive events, then [tex]\( p(A \cup B) = 1 \)[/tex]. This means the probability that either [tex]\( A \)[/tex] or [tex]\( B \)[/tex] or both occur is 1.
2. Given Probabilities:
- [tex]\( p(A) = 0.7 \)[/tex]
- [tex]\( p(B) = 0.4 \)[/tex]
3. Applying the Principle for Exhaustive Events:
- Since [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are exhaustive:
[tex]\[
p(A \cup B) = 1
\][/tex]
- We use the formula for the probability of the union of two events:
[tex]\[
p(A \cup B) = p(A) + p(B) - p(A \cap B)
\][/tex]
4. Substituting the Given Values:
- Substituting [tex]\( p(A \cup B) = 1 \)[/tex]:
[tex]\[
1 = p(A) + p(B) - p(A \cap B)
\][/tex]
[tex]\[
1 = 0.7 + 0.4 - p(A \cap B)
\][/tex]
5. Solving for [tex]\( p(A \cap B) \)[/tex]:
- Rearrange the equation to solve for [tex]\( p(A \cap B) \)[/tex]:
[tex]\[
1 = 1.1 - p(A \cap B)
\][/tex]
[tex]\[
p(A \cap B) = 1.1 - 1
\][/tex]
[tex]\[
p(A \cap B) = 0.1
\][/tex]
6. Conclusion:
- The probability that both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occur is [tex]\( 0.1 \)[/tex].
Therefore, the answer is:
[tex]\[ \boxed{0.1} \][/tex]
From the given options:
a) 0.3
b) 0.1
c) 1
d) can't be determined
The correct answer is:
b) 0.1
