# Probability- significant difference between $(A' \cap B)$ & $(A' \cup B)$ or $(A \cap B')$ & $(A \cup B')$?

Venn Diagrams in general! I've honestly been struggling with these kinds of questions for HOURS, I still don't get it. I went on the internet, researched to find an answer but only stumbled upon a site that shows me some questions with their answers but not the solutions on how they got the answer.

Please help me find a way to get the answers. I'm really confused, not sure what to do. Did my best. Here are the questions and thank you guys so much in advance!

$4$) The events $A$ and $B$ are such that $$p(A) = 0.5,\quad p(B) = 0.7,\quad p(A \cap B) = 0.2.$$ Find $p(A' \cap B)$ --- ans: $0.5$ (I know how they got $0.5$ ($=1-p(A)$), but I was confused with the difference of this question compared to number $5$. I tried the same method for number $5$ ($1-p(B)$), but I didn't get the right answer.)

What's the significant difference between $(A' \cap B)$ & $(A' \cup B)$ or $(A \cap B')$ & $(A \cup B')$ ???

5) The events $A$ and $B$ are such that $$p(A) = 0.35,\quad p(B) = 0.5, \quad p(A \cap B) = 0.15.$$ Find $p(A \cup B’)$ ans: $0.65$.

6) The events $A$ and $B$ are such that $$p(A) = 0.45,\quad p(B) = 0.7,\quad p(A \cap B) = 0.20.$$ Find
(a) $p(A’ \cap B’)$ --ans: $0.05$
(b) $p(A \cap B)’)$ --ans: $0.80$

Those are the questions I got from the internet. When I figure out how to do these I'll be able to answer my own questions from school. Thanks so much for the help!

This is if you want to help me further: My school questions that I have been struggling with for hours:

1) If $p(A) = 0.6, p(B) = 0.3$ and $p(A \cap B) = 0.2$ find

$p(A \cap B’)$ --i don't have an answer to this :( I need a solution though

2) $A$ and $B$ are two events such that $p(A)= p, p(B)=2p$ and $p(A \cap B) =p^2$ Use a venn diagram to help you find the following:
a. $p(A' \cup B)$
b. $p(A' \cap B' )$

Last question--3) A card is drawn at random from a standard deck of $52$ playing cards. Find the probability that the card is a club or spade.

The struggle is real.

Q4: $$p(A) = 0.5,\quad p(B) = 0.7,\quad p(A \cap B) = 0.2.$$ Find $p(A' \cap B)$ = $p(B) - P(A \cap B)$ = 0.7-0.2 = 0.5

Q5:$$p(A) = 0.35,\quad p(B) = 0.5, \quad p(A \cap B) = 0.15.$$

$p(A \cup B)’)$ = 1-$p(A \cup B))$ = $1-(0.35+0.5-0.15)$ = $0.3$

$p(A \cup B’)$ = $p(A)+p(A \cup B)’)$ = $0.35+0.3$ = $0.65$

Q6:

(a) $p(A’ \cap B’)$ = $1-p(A \cup B))$ = $1-.45-.7+.2$ = $0.05$

(b) $p(A \cap B)’)$ = $1-p(A \cap B))$ = $1-.2$ = $0.8$

School Homework:

$p(A \cap B’)$ = $1-p(B) = 1-0.3 = 0.7$

$p(A' \cup B)$ = $2p-p^2 +1 - (p+2p-p^2)$ = $1-p$

$p(A’ \cap B’)$ = $1-p(A \cup B)$ = ($1-(p+2p-p^{2}$) = $1-3p+p^{2}$

P(the card is a club or spade) = P(the card is club) + P(the card is spade)

= $\dfrac{{13\choose1} + {13\choose1}}{{52\choose1}}$

For question 4, notice that since $A' \cap B$ and $A \cap B$ form a partition of $B$, we have: $$p(A' \cap B) = p(B) - p(A \cap B) = 0.7 - 0.2 = 0.5$$

For question 5, notice that since $A' \cap B$ and $A \cap B$ form a partition of $B$, we have: $$p(A' \cap B) = p(B) - p(A \cap B) = 0.5 - 0.15 = 0.35$$ Thus, by DeMorgan's Law, we have: $$p(A \cup B') = 1 - p((A \cup B')') = 1 - p(A' \cap B) = 1 - 0.35 = 0.65$$

For question 6, notice that by the principle of inclusion-exclusion, we have: $$p(A \cup B) = p(A) + p(B) - p(A \cap B) = 0.45 + 0.7 - 0.20 = 0.95$$ Thus, by DeMorgan's Law, we have: $$p(A' \cap B') = p((A \cup B)') = 1 - p(A \cup B) = 1 - 0.95 = 0.05$$ Furthermore, note that: $$p((A \cap B)') = 1 - P(A \cap B) = 1 - 0.20 = 0.80$$

Hopefully you can do the rest on your own now.