Question

Please explain me why
If P(A) = 0.4
P(B) = 0.5
P(A and B) = 0.3
Find
a)P(A or B)
b) P (A or B')
c) P(A ' or B')
I want to make clear about.

Answer 1

P(A') = 0.6
P(B') =0.5
\(P(A\cup B) = P(A) + P(B) -P(A\cap B)= 0.4+0.5-0.3 = 0.6\)
\(P(A \cup B') = P(A) +P(B') -P (A\cap B') \)

Answer 2

How to find P(A and B') without using Venn?

Answer 3

\[P(A)=P(A\cap B)+P(A\cap B')\implies P(A\cap B')=P(A)-P(A\cap B)\]

The first formula there is actually an important (but sometimes subtle) formula. It's basically called the "Partition rule" or "Law of Total probability"

Answer 4

Thank you, I got it
now, the last part: \((A\cap B) ' = (A'\cup B') \)
and \(P (A\cap B)' = 1-P(A \cap B) = 1- 0.3= 0.7=P(A' \cup B')\)

Answer 5

If you are interested, The formula can be generalized. If \(B_1 \cup B_2 \cup \cdots \cup B_n\) form a partition of the sample space (That is \(B_i\cap B_j=\emptyset\) for \(i\ne j\)), i.e the \(B_i\)'s are mutually exclusive, then \[P(A)=\sum_{i=1}^nP(A \cap B_i) \] And you can extened this with conditonal probabilities:
\[ P(A)=\sum_{i=1}^nP(A|B_i)P(B_i)\]

Answer 6

You are missing some areas in your Venn diagram