Probability of independent events

[desmond iking]

i have drawn the venn diagram for the question... i am not sure it's correct or not. the question states both are independent events. i don't understand the working in part b ... for me , my ans would be p(A) +p(B).. correct me if i am wrong. thanks in advance.

 

[Simon Bridge]

Set A should be the set of all products with defect A
Similarly for set B.
Now can you see why the sets intersect?

What does that look like on a venn diagram? (hint: not like you did).

Independent events are those whose probability is not affected by the occurrence of the other.
i.e. where P(A|B)=P(A), P(B|A)=P(B)

Since: P(A|B)=P(AnB)/P(B)
This means that for independent events, P(AnB)=P(A)P(B).

(using AnB = "A intersection B")

 

[desmond iking]

Set A should be the set of all products with defect A
Similarly for set B.
Now can you see why the sets intersect?

What does that look like on a venn diagram? (hint: not like you did).

Independent events are those whose probability is not affected by the occurrence of the other.
i.e. where P(A|B)=P(A), P(B|A)=P(B)

Since: P(A|B)=P(AnB)/P(B)
This means that for independent events, P(AnB)=P(A)P(B).

(using AnB = "A intersection B")

the venn diagram look like this?

 

[micromass]

the venn diagram look like this?

Correct. Do you see now why your answer was wrong? You essentially did the following:

\mathbb{P}(A\cup B) = \mathbb{P}(A) + \mathbb{P}(B)

which is only allowed if $A$ and $B$ do not intersect. But $A$ and $B$ do intersect here, so you cannot apply this formula!

 

[FactChecker]

To be a little more specific about independent events -- If events A and B are independent, the probability of event A happening is the same whether B occurs or not. And the converse is true. One way to show that in a Venn diagram is to draw A as a horizontal slice of the whole diagram, and B as a vertical slice of the whole diagram. That way the percent of A within B is the same as the percent of A within the entire diagram (and vice versa). The first diagram in the attachment shows independent events and the second diagram shows dependent events