Exercise · ZW-PROBABILITY-MEASURE-009 · ★★★ Advanced
If \(P(A)=0.6\) and \(P(B)=0.7\), what is the smallest possible value of \(P(A\cap B)\)?
Choose one option.
Show answer
- C. \(0.3\)
Explanation
Since \(P(A\cup B)\le 1\), \(P(A\cap B)=P(A)+P(B)-P(A\cup B)\ge 1.3-1=0.3\). The value \(0.42\) is the product, which requires independence; it is not the minimum.
Review the concept: Probability measure
A probability measure \(P\) is a function that assigns to each event \(A\) a number between 0 and 1, equals 1 on the sample space \(\Omega\) and adds over mutually exclusive events.
\[P(\Omega)=1,\qquad P\Big(\bigcup_i A_i\Big)=\sum_i P(A_i)\]
More practice problems: probability measure
- ZW-PROBABILITY-MEASURE-001 A fair die has P(\k\)=16 for each face. What is P(\2,4,6\)?
- ZW-PROBABILITY-MEASURE-002 If A and B are mutually exclusive with P(A)=0.3 and P(B)=0.5, what is P(A∪ B)?
- ZW-PROBABILITY-MEASURE-003 Let Omega=\a,b,c\. Which assignment for (P(a),P(b),P(c)) defines a probability measure?
- ZW-PROBABILITY-MEASURE-004 If P(A)=0.35, what is P(A^c)?
- ZW-PROBABILITY-MEASURE-005 Which of these values cannot be the probability of any event?
- ZW-PROBABILITY-MEASURE-006 If P(A)=0.4, P(B)=0.5 and P(A∩ B)=0.2, what is P(A∪ B)?
- ZW-PROBABILITY-MEASURE-007 If A⊆ B, P(A)=0.3 and P(B)=0.5, what is P(Bsetminus A)?
- ZW-PROBABILITY-MEASURE-008 It is known that P(A)=0.5, P(B)=0.6 and P(A∪ B)=0.9. What is P(A∩ B)?