Show that
P(A∪B) = P(A) + P(B)−P(A∩B)
If B⊂A, show that P(A)=P(B)+P(A∩BC)
Show that the condition A = B is not a necessary one for P(A) ∪ P(B) = P(A ∪ B) by finding examples of sets A and B such that P(A) ∪ P(B) = P(A ∪ B) but A does not equal B.
For any events A and B with P(B)>0, show that P(A|B) + P(A'|B) = 1. How do you draw a venn diagram to solve/illustrate this problem?
Exercise 2. The Principle of Inclusion-Exclusion Show that P(A∪B) = P(A) + P(B)−P(A∩B) (1.4) By writing A∪B∪C as (A∪B)∪C, extend the result to three sets: P(A∪B∪C) = P(A)+P(B)+P(C)−P(A∩B)−P(A∩C)−P(B∩C)+P(A∩B∩C)
m u cSIUNI II that order! If A and B are two mutually exclusive events show that P(A/B) = - P(A) P-P(B)
Let P(A) = 0.4 P(B) = 0.5 P(A|B) = 0.2 (Please show working). If the events a and b are independent, calculate the P(A and B) If the events a and b are not independent, calculate the P(A and B) If the events a and b are mutually exclusive, calculate the P(A or B)
5. If P(AB)-1, show that P(B (i) P(A n B n C) (ii) P(C|A B)-P(CB); and n C) for any event C (iii) P(A n CB) P(CB).
Show via probability manipulation that the following is true: P(A∩B∩C) = P(A|B∩C) x P(B|C) x P(C)
Prove the following statements. Show your working. (i) If P(A|B,C) = P(B|A,C), then P(A|C) = P(BIC) (ii) If P(A|B,C) = P(A), then P(B,C|A) = P(B,C) (iii) If P(A, B|C) = P(A|C) * P(BIC), then P(A|B,C) = P(AC)
Problem 3. Show the formula P((An B)U(A n B))- P(A) +P(B)-2P(AnB), which givgs the probability that exactly one of the events A and B will occur. [Compare with the formula P(AU B) P(A) P(B) - P(AnB), which gives the probability that at least one of the events A and B will occur.]