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# Probability

### Example 1: using the probability rules

An experiment consists of rolling a single die. Find the probability that the number appearing uppermost is:

a) a $$5$$ or a $$6$$?
b) a number less than $$5$$?
c) an $$8$$?
d) an even number or a number less than $$4$$?

Solution:

An experiment consists of rolling a single die. The probability distribution will be:

 $$x$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$P(x)$$ $$\displaystyle \frac{1}{6}$$ $$\displaystyle \frac{1}{6}$$ $$\displaystyle \frac{1}{6}$$ $$\displaystyle \frac{1}{6}$$ $$\displaystyle \frac{1}{6}$$ $$\displaystyle \frac{1}{6}$$

a) \begin{eqnarray*}
&& P(x = 5 \mbox{ or } x = 6) \\
&=& P(x = 5) + P(x = 6) \\
&=& \frac{1}{6} + \frac{1}{6} \\
&=& \frac{1}{3}
\end{eqnarray*}

b) \begin{eqnarray*}
&& P(x < 5) \\
&=& P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) \\
&=& \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} \\
&=& \frac{2}{3}
\end{eqnarray*}

c) $P(x=8)=0$

d) \begin{eqnarray*}
&&P(x \mbox{ is even, or } x < 4) \\
&=& P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) \\