Easy Stats/Set Theory Question

[Tracker]

This is a question I have worked on in Stats and my answer does not line up with the solution manuals.

 

[math] P(A' \cap B' \cap C) = P© - P(A \cap C) - P(B \cap C)

+ P(A \cap B \cap C) [/math]

 

My answer there is a negative before [math] P(A \cap B \cap C) [/math]

[Samar]

I'm afraid your solution manuals is correct.

This is the Venn diagram of your problem.

What you want to do is to say P© without that region of intersection with A alone,B alone + the region of intersection of three circles.Right?

but when you say [math]P(A \cap C)[/math] you refer to the region in which A intersects C alone + that one in which the three circles intersect with each other.

And when you say [math] P(B \cap C) [/math] you also refer to the region in which B intersects with C alone + that one in which the three circles intersect.

If you added both values to each other with negative before them you'd be saying:It's the probability of C without regions of intersecion with A alone + B alone + 2 times the region of intersection of three circles.

That's why you compensate that by adding [math]P(A \cap B \cap C)[/math] with positive before it at the end of the equation.

I hope this helped