Probability of Independent events

[Shawna]

Hey guys, I have trouble understanding how to know if 2 events are independant or dependant.

For example, if P(A∪B)=0.95 ; P(A)=0.45 ; P(B)=0.5

Based on: P(A∪B)= P(A)+P(B) These would appear to be 2 independent events, with A∩B=Ø

However, what if we decide to draw a Venn diagram that looks like this?

 

[DrPhil]

Hey guys, I have trouble understanding how to know if 2 events are independant or dependant.

For example, if P(A∪B)=0.95 ; P(A)=0.45 ; P(B)=0.5

Based on: P(A∪B)= P(A)+P(B) These would appear to be 2 independent events, with A∩B=Ø

However, what if we decide to draw a Venn diagram that looks like this:
View attachment 3027
P(A∪B)= P(A)+P(B)-A∩B ->
0.95= 0.45+0.5-0.05 utilizing the given data, this can make sense too no? Making these events dependent.

What am I missing here?

The situation you describe first, with A∩B=Ø, is two mutually exclusive events. These are not independent, because the truth of either makes the other false.

Independence means the probability of A is the same, no matter what B is. Writing in terms of conditional probabilities,

P(A) = P(A | B) = P(A | notB}

P(B) = P(B | A) = P(B | notA)

Can you use the Venn diagram to determine whether those statements are true?

 

[Shawna]

Nevermind, realized the diagram would sum to 0.9

What a blonde moment

 

[JeffM]

Hey guys, I have trouble understanding how to know if 2 events are independant or dependant.

For example, if P(A∪B)=0.95 ; P(A)=0.45 ; P(B)=0.5

Based on: P(A∪B)= P(A)+P(B) These would appear to be 2 independent events, with A∩B=Ø

However, what if we decide to draw a Venn diagram that looks like this?
View attachment 3027

Your Venn diagram does not describe the situation specified.

The probabilities described in your diagram are

\(\displaystyle P(A \bigcap \neg B) = 0.4\)

\(\displaystyle P(B \bigcap \neg A) = 0.45\)

\(\displaystyle P(A \bigcap B) = 0.05\)

So \(\displaystyle P(A) = P(A \bigcap \neg B) + P(A \bigcap B) = 0.4 + 0.05 = 0.45.\) This fits your initial description.

And \(\displaystyle P(B) = P(B \bigcap \neg A) + P(B \bigcap A) = 0.45 + 0.05 = 0.5.\) This also fits your initial description.

But \(\displaystyle P(A \bigcup B) = P(A \bigcap \neg B) + P(A \bigcap B) + P(B \bigcap \neg A) = 0.4 + 0.05 + 0.45 = 0.9 \ne 0.95.\) This does not fit your initial description.

 

[pka]

Hey guys, I have trouble understanding how to know if 2 events are independant or dependant.
For example, if P(A∪B)=0.95 ; P(A)=0.45 ; P(B)=0.5
Based on: P(A∪B)= P(A)+P(B) These would appear to be 2 independent events, with A∩B=Ø
However, what if we decide to draw a Venn diagram that looks like this?
View attachment 3027

First, I don't fully understand the confusion here.

You should know that \(\displaystyle \mathcal{P}(A\cup B)=\mathcal{P}(A)+\mathcal{P}( B)-\mathcal{P}(A\cap B)\) is always true.

Therefore, if \(\displaystyle \mathcal{P}(A)>0~\&~\mathcal{P}(B)>0\) and \(\displaystyle \mathcal{P}(A\cup B)=\mathcal{P}(A)+\mathcal{P}( B)\) then it must be the case that \(\displaystyle A\cap B=\emptyset\) and those events cannot be independent.

Recall that if two events are independent then \(\displaystyle \mathcal{P}(A\cap B)=\mathcal{P}(A)\cdot\mathcal{P}( B)\).

Does that even begin to address you concerns?

 

[Shawna]

Sorry,
I meant to say 'exclusive' instead of 'independant'. (English isn't my native language). My bad.
Thanks for the help, you did clarify it to me well Pka.