Hello, amul28!
An interesting problem . . . It takes a bit of Thinking.
What does minimum and maximum values of a conditional probability )
mean?
When =0.3,\; P(A\cap B)=0.1)
Bayes' Theorem: .  \:=\:\dfrac{P(A \cap B)}{P(B)} )
And we have: .  \;=\;\dfrac{0.1}{P(B)} )
.[1]
What is the range of values for )
?
The Venn diagram looks like this:
Code:
* - - - - - - - - - *
| |
| *-------* |
| | A | |
| |0.2*---+---* |
| | | | | |
| | |0.1| | |
| *---+---* | |
| | B | |
| *-------* |
| |
* - - - - - - - - - *
We are told that: .  \:=\:0.1)
That goes in the region common to circles 
and 
We are told that: .  = 0.3)
The 0.3 goes in the entire -circle.
That leaves 0.2 for the )
region,
. . the region that is in , but not in
What values can go in the region )
?
. . The value can range from 
to 
.
Hence, can range from 0.1 to 0.8.
Substitute into [1].
 = 0.8,\text{ then: }\;P(A|B) \:=\:\dfrac{0.1}{0.8} \:=\:0.125\;\text{ minimum})
 = 0.1,\text{ then: }\;P(A|B) \:=\:\dfrac{0.1}{0.1} \:=\:1.0 \quad\text{ maximum})
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conditional mini max
