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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) \\
&& \quad {}+ P(x=6) \\
&=& \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} \\
&=& \frac{5}{6}
\end{eqnarray*}