ONE MORE STEP TOWARD BETTER TOMORROW : chapter-14 Worked Out Examples

Thursday 7 August 2014

chapter-14 Worked Out Examples

Example: 11

For two events $A$ and $B$ , find the probability that exactly one of the two events occur.

Solution: 11

This can happen in two ways:

	Way	Set of Outcomes
	$A$ occurs, $B$ doesn’t	$A \cap \bar B$
	$B$ occurs, $A$ doesn’t	$\bar A \cap B$

Note that the two ways are $ME$ . Thus, the required probability is $P\left( {A \cap \bar B} \right) + P\left( {\bar A \cap B} \right)$ , which from the second relation becomes

$P(A) - P\left( {A \cap B} \right) + P\left( B \right) - P\left( {A \cap B} \right)$ $= P\left( A \right) + P\left( B \right) - 2P\left( {A \cap B} \right)$

Justify this last expression using a Venn diagram, by shading the area it represents.

Example: 12

For two events $A$ and $B$ , show that

$P\left( B \right) = P\left( A \right) \cdot P\left( {B/A} \right) + P\left( {\bar A} \right) \cdot P\left( {B/\bar A} \right)$

Solution: 12

This is straightforward, since

	$P\left( A \right) \cdot P\left( {B/A} \right) = P\left( {A \cap B} \right)$	recall the discussion in the previous section
	and $P\left( {\bar A} \right) \cdot P\left( {B/\bar A} \right) = P\left( {\bar A \cap B} \right)$