how to do probability questions make table edition

worked example: if $\text{Pr}(A)=0.4$, $\text{Pr}(B)=0.5$ and $\text{Pr}(A\cap B)=0.2$, find

• $\text{Pr}(A\cup B)$
  make a table

	A	A’
B	0.2	0.3	0.5
B’	0.2	0.3	0.5
	0.4	0.6	1

$\text{Pr}(A\cup B)=0.7$

• $\text{Pr}(A|B)$
  $\text{Pr}(A|B)=\frac{0.2}{0.5}=\frac{2}{5}$

worked example: if $\text{Pr}(A)=0.44$, $\text{Pr}(B)=0.16$ and $\text{Pr}(A\cup B)=0.53$, find

• $\text{Pr}(A\cap B)$
  make a table
  $\text{Pr}(A\cup B)=0.53$ means that $\text{Pr}(A^{\prime }\cap B^{\prime })=0.47$

	A	A’
B	0.07	0.09	0.16
B’	0.37	0.47	0.84
	0.44	0.56	1

$\text{Pr}(A\cap B)=0.07$

• $\text{Pr}(B|A)$
  $\text{Pr}(B|A)=\frac{0.07}{0.44}=\frac{7}{44}$

worked example: if A and B are mutually exclusive events and $\text{Pr}(A)=0.38$ and $\text{Pr}(B)=0.51$, find $\text{Pr}(A\cup B)$
if A and B are mutually exclusive, $\text{Pr}(A\cap B)=0$
make a table

	A	A’
B	0	0.51	0.51
B’	0.38	0.11	0.49
	0.38	0.62	1

$\text{Pr}(A\cup B)=0.89$

worked example: if A and B independent events and $\text{Pr}(A)=0.3$ and $\text{Pr}(B)=0.4$, find $\text{Pr}(A\cup B)$
if A and B are independent, $\text{Pr}(A\cap B)=\text{Pr}(A)\cdot \text{Pr}(B)$
make a table

	A	A’
B	0.12	0.28	0.4
B’	0.18	0.42	0.6
	0.3	0.7	1

$\text{Pr}(A\cup B)=0.58$

worked example: if $\text{Pr}(A)=2\cdot \text{Pr}(B)$ and $\text{Pr}(A\cap B)=0.2$ and $\text{Pr}(A\cup B)=0.7$, find $\text{Pr}(A)$
make a table
$\text{Pr}(A\cup B)=0.7$ means that $\text{Pr}(A^{\prime }\cap B^{\prime })=0.3$

	A	A’
B	0.2
B’		0.3
			1

we have to use some variables here

	A	A’
B	0.2	p	n
B’	o	0.3
	m		1

some equations

$m=2n$
$0.2+o+p=0.7$
$0.2+o=m$
$0.2+p=n$

$0.2+o+0.2+p=0.9$
$0.2+o+0.2+p=3n$
$n=0.3$

phew that was hard

	A	A’
B	0.2	0.1	0.3
B’	0.4	0.3	0.7
	0.6	0.4	1

$\text{Pr}(A)=0.6$