# A fair die is rolled. Consider events E = {1,3,5}, F = {2,3} and G = {2,3,4,5}. Calculate $P \large ( \large \frac{ E \cup F}{G})$ and $P \large ( \large \frac{ E \cap F}{G})$

Toolbox:
• In any problem that involves tossing a coin or a die and determing the outcomes, we can write down the sample space and cout the number of favorable outcomes. Then we find P(E), P(F), P(E$\cap$F) using the set of outcomes.
• Given P(E), P(F), P(E $\cap$ F), P(E/F) $= \large \frac{P(E \;\cap \;F)}{P(F)}$
• $P \large ( \large \frac{ E \cup F}{G})$ = $\large \frac {P ((E \cup F) \cap G)}{P(G)}$
• $P \large ( \large \frac{ E \cap F}{G})$ = $\large \frac {P ((E \cap F) \cap G)}{P(G)}$
Given that a fair die is rolled, number of total outcomes = 6.
Given E: {1, 3, 5} $\rightarrow P(E) = \large \frac{\text{Number of favorable outcomes in E}}{\text{Total number of outcomes in S}} = \frac{3}{6} = \frac{1}{2}$
Given F: {2,3} $\rightarrow P(F) = \large \frac{\text{Number of favorable outcomes in F}}{\text{Total number of outcomes in S}} = \frac{2}{6} = \frac{1}{3}$
Given G: {2,3,4,5} $\rightarrow P(G) = \large \frac{\text{Number of favorable outcomes in G}}{\text{Total number of outcomes in S}} = \frac{4}{6} = \frac{2}{3}$
It also follows that we can calculate the various "Union" and "Intersection" of the sets and their probabilities as follows:
$\Rightarrow E \cup F = {1,2,3,5} \rightarrow P( E \cup F) = \large \frac{4}{6} = \frac{2}{3}$
$\Rightarrow E \cap F = {3} \rightarrow P( E \cap F) = \large \frac{1}{6}$
$\Rightarrow E \cup G = {1,2,3,4,5} \rightarrow P( E \cup G) = \large \frac{5}{6}$
$\Rightarrow E \cap G = {3,5} \rightarrow P( E \cap G) = \large \frac{2}{6} = \frac{1}{3}$
$\Rightarrow (E \cup F) \cap G = {2,3,5} \rightarrow P ((E \cup F) \cap G) = \large \frac{3}{6} = \frac{1}{2}$
$\Rightarrow (E \cap F) \cap G = {3} \rightarrow P ((E \cap F) \cap G) = \large \frac{1}{6}$
Given $P \large ( \large \frac{ E \cup F}{G})$ = $\large \frac {P ((E \cup F) \cap G)}{P(G)}$ and $P \large ( \large \frac{ E \cap F}{G})$ = $\large \frac {P ((E \cap F) \cap G)}{P(G)}$
$P \large ( \large \frac{ E \cup F}{G})$ = $\Large \frac{\Large \frac{1}{2}} {\frac{2}{3}}$ = $\large \frac{3}{4}$
$P \large ( \large \frac{ E \cap F}{G})$ = $\Large \frac{\Large \frac{1}{6}} {\frac{2}{3}}$ = $\large \frac{1}{4}$