# Math Help - conditional mini max

1. ## conditional mini max

what does minimum and maximum values of a conditional probability P(A/B) mean?
When P(A)=0.3, P(A ∩ B)=0.1

2. Hello, amul28!

An interesting problem . . . It takes a bit of Thinking.

What does minimum and maximum values of a conditional probability $P(A|B)$ mean?
When $P(A)=0.3,\; P(A\cap B)=0.1$

Bayes' Theorem: . $P(A|B) \:=\:\dfrac{P(A \cap B)}{P(B)}$

And we have: . $P(A|B) \;=\;\dfrac{0.1}{P(B)}$ .[1]

What is the range of values for $P(B)$?

The Venn diagram looks like this:

Code:

* - - - - - - - - - *
|                   |
|   *-------*       |
|   | A     |       |
|   |0.2*---+---*   |
|   |   |   |   |   |
|   |   |0.1|   |   |
|   *---+---*   |   |
|       |     B |   |
|       *-------*   |
|                   |
* - - - - - - - - - *

We are told that: . $P(A \cap B) \:=\:0.1$
That goes in the region common to circles $\,A$ and $\,B.$

We are told that: . $P(A) = 0.3$
The 0.3 goes in the entire $\,A$-circle.
That leaves 0.2 for the $(A \cap B')$ region,
. . the region that is in $\,A$, but not in $\,B.$

What values can go in the region $(B \cap A')$ ?
. . The value can range from $0.0$ to $0.7$.
Hence, $P(B)$ can range from 0.1 to 0.8.

Substitute into [1].

$\text{If }P(B) = 0.8,\text{ then: }\;P(A|B) \:=\:\dfrac{0.1}{0.8} \:=\:0.125\;\text{ minimum}$

$\text{If }P(B) = 0.1,\text{ then: }\;P(A|B) \:=\:\dfrac{0.1}{0.1} \:=\:1.0 \quad\text{ maximum}$

3. thank u.

even i found the same way but got messed up near 0.8 now its done.