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# Probability

### Example 3: using the probability rules

A treasure chest contains $$30$$ diamonds, $$10$$ gold coins, $$20$$ sapphires and $$5$$ silver medallions. If an item is selected at random, find the probability that it is:

a) a gold coin?
b) not a sapphire?
c) either a gold coin or a silver medallion?
d) neither a gold coin nor a silver medallion?

Solution:

a) \begin{eqnarray*}
&&P(\mbox{Gold}) \\
&=& \frac{\mbox{Number of Gold coins}}{\mbox{Total number of items in the chest}} \\
&=& \frac{10}{65} \\
&=& \frac{2}{13} \approx 0.1538
\end{eqnarray*}

b) \begin{eqnarray*}
&&P(\mbox{Not a Sapphire}) \\
&=& \frac{65 - 20}{65} \\
&=& \frac{45}{65} \\
&=& \frac{9}{13} \approx 0.6923
\end{eqnarray*}

c) \begin{eqnarray*}
&&P(\mbox{Gold or Silver}) \\
&=& \frac{10}{65} + \frac{5}{65} \\
&=& \frac{15}{65} \\
&=& \frac{3}{13} \approx 0.2308
\end{eqnarray*}

d) \begin{eqnarray*}
&&P(\mbox{Not gold or silver}) \\
&=& 1 - \frac{3}{13} \\
&=& \frac{10}{13} \approx 0.7692
\end{eqnarray*}