You are given

• [math]\operatorname{P}[A \cup B] = 0.7[/math]
• [math]\operatorname{P}[A \cup B'] = 0.9[/math]

Calculate [math]\operatorname{P}[A] [/math].

• 0.2
• 0.3
• 0.4
• 0.6
• 0.8

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: D

First note

[[math]] \operatorname{P}[ A ∪ B ] = \operatorname{P}[ A] + \operatorname{P}[ B ] − \operatorname{P}[ A ∩ B ], \, \operatorname{P}[ A ∪ B '] = \operatorname{P}[ A] + \operatorname{P}[ B '] − \operatorname{P}[ A ∩ B ']. [[/math]]

Then add these two equations to get

[[math]] \begin{align*} \operatorname{P}[ A ∪ B ] + \operatorname{P}[ A ∪ B '] &= 2 \operatorname{P}[ A] + ( \operatorname{P}[ B ] + \operatorname{P}[ B '] ) − ( \operatorname{P}[ A ∩ B ] + \operatorname{P}[ A ∩ B '] ) \\ 0.7 + 0.9 &= 2 \operatorname{P}[ A] + 1 − P ⎡⎣( A ∩ B ) ∪ ( A ∩ B ') ⎤⎦ \\ 1.6 &= 2 \operatorname{P}[ A] + 1 − \operatorname{P}[ A] \\ \end{align*} [[/math]]

which implies that [math]\operatorname{P}[A] = 0.6 [/math].

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.