Respuesta :
Answer:
At least one of the events A and B occurs:
(a) 0.5
(b) 0.44
All of the events A, B, C occur:
(a) 0.024
(b) 0
Step-by-step explanation:
Given:
P(A) = 0.2, P(B) = 0.3, P(C) = 0.4
At least one of the events A and B occurs:
(a) If A and B are mutually exclusive events, then their intersection is 0.
Probability of at least one of them occurring means either of the two occurs or the union of the two events. Therefore,
[tex]P(A\ or\ B)=P(A)+P(B)-P(A\cap B)\\P(A\ or\ B)=0.2+0.3-0=0.5[/tex]
(b) If A and B are independent events, then their intersection is equal to the product of their individual probabilities. Therefore,
[tex]P(A\ or\ B)=P(A)+P(B)-P(A\cap B)\\P(A\ or\ B)=P(A)+P(B)+P(A)P(B)\\P(A\ or\ B)=0.2+0.3-(0.2)(0.3)=0.5-0.06=0.44[/tex]
All of the events A, B, C occur together:
(a) All of the events occurring together means the intersection of all the events.
If A, B, and C are independent events, then their intersection is equal to the product of their individual probabilities. Therefore,
[tex]P(A\cap B\cap C)=P(A)\cdot P(B)\cdot P(C)\\P(A\cap B\cap C)=0.2\times 0.3\times 0.4=0.024[/tex]
(b) If A, B, and C are mutually exclusive events means events A, B, and C can't happen at the same time. Therefore, their intersection is 0.
[tex]P(A\cap B\cap\ C)=0[/tex]
