Given the following probabilities: $p\left(a\right)=0.3$, $p\left(b\right)=0.2$, and $p(a\wedge b)=0.4$, which of the following statements is true?

• A. $p(a\vee b)=0.1$
• B. $p(a\wedge b)\le p\left(b\right)$
• C. $p(a\wedge b)\le p\left(a\right)$
• D. At least one of the given probabilities is not possible for valid events.

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the given probabilities and their meanings. We have \( p(a) = 0.3 \), \( p(b) = 0.2 \), and \( p(a \cap b) = 0.4 \). Here, \( p(a \cap b) \) represents the probability that both events \( a \) and \( b \) occur simultaneously.
2. Step 2: Recall the fundamental property of probabilities for intersections: the probability of the intersection of two events cannot be greater than the probability of either event individually. Mathematically, this means \( p(a \cap b) \leq p(a) \) and \( p(a \cap b) \leq p(b) \).
3. Step 3: Compare the given intersection probability \( p(a \cap b) = 0.4 \) with the individual probabilities \( p(a) = 0.3 \) and \( p(b) = 0.2 \). Since \( 0.4 > 0.3 \) and \( 0.4 > 0.2 \), this violates the property mentioned in Step 2.
4. Step 4: Because the intersection probability is greater than the individual probabilities, at least one of the given probabilities must be incorrect or impossible for valid events. This means the data provided is inconsistent with the axioms of probability.
5. Step 5: To further verify, you could check the union probability using the formula \( p(a \cup b) = p(a) + p(b) - p(a \cap b) \) and ensure it lies between 0 and 1. However, since the intersection is already invalid, the problem is identified at this step.