Respuesta :
Answer:
Case(a): [tex]p[b]=\frac{1}{3}p[a][/tex]
Case(b): [tex]p[b]=0[/tex]
Case(c): [tex]p[b]=\frac{1}{2}p[a\cap b][/tex]
Step-by-step explanation:
Given
(a) events a and b are a partition and p[a] = 3p[b].
(b) for events a and b, p[a ∪ b] = p[a] and p[a ∩ b] = 0.
(c) for events a and b, p[a ∪ b] = p[a]− p[b].
we have to find the p[b] in each case:
Case (a): events a and b are a partition and p[a] = 3p[b].
gives [tex]p[b]=\frac{1}{3}p[a][/tex]
Case (b): for events a and b, p[a ∪ b] = p[a] and p[a ∩ b] = 0.
[tex]p[a\cup b]=p[a][/tex] ⇒[tex]p[a]+p[b]-p[a\cap b]=p(a)[/tex] ⇒ [tex]p[b]=0[/tex] ∵ p[a ∩ b] = 0.
Case(3): for events a and b, p[a ∪ b] = p[a]− p[b].
p[a ∪ b] = p[a]− p[b]
⇒ [tex]p[a]+p[b]-p[a\cap b]=p[a]-p[b][/tex]
⇒ [tex]2p[b]=p[a\cap b][/tex]
⇒ [tex]p[b]=\frac{1}{2}p[a\cap b][/tex]
Answer:
(a) [tex]p[b]=\frac{1}{3}\times p[a][/tex].
(b) p[b]=0
(c) [tex]p[b]=\frac{1}{2}\times p[a\cap b][/tex]
Step-by-step explanation:
(a)
[tex]p[a]=3p[b][/tex]
Divide both sides by 3.
[tex]\frac{1}{3}\times p[a]=p[b][/tex]
Therefore [tex]p[b]=\frac{1}{3}\times p[a][/tex].
(b)
[tex]p[a\cup b]=p[a][/tex]
[tex]p[a]+p[b]-p[a\cap b]=p[a][/tex] [tex][\because P(A\cup B)=P(A)+P(B)-P(A\cap B)][/tex]
[tex]p[a]+p[b]-0=p[a][/tex] [tex][\because p[a\cap b]=0][/tex]
[tex]p[b]=p[a]-p[a][/tex]
[tex]p[b]=0[/tex]
Therefore [tex]p[b]=0[/tex].
(c)
[tex]p[a\cup b]=p[a]-p[b][/tex]
[tex]p[a]+p[b]-p[a\cap b]=p[a]-p[b][/tex] [tex][\because P(A\cup B)=P(A)+P(B)-P(A\cap B)][/tex]
[tex]p[b]-p[a\cap b]+p[b]=0[/tex]
[tex]2p[b]=p[a\cap b][/tex]
[tex]p[b]=\frac{1}{2}\times p[a\cap b][/tex]
Therefore [tex]p[b]=\frac{1}{2}\times p[a\cap b][/tex].
