Math – Exercise 3 (Stochastic)

Given

Two dice (cubes, 6 faces, 1 to 6) are getting rolled.

Find

The probability that the sum is smaller then 5 or greater then 8.

Solution

\(
\begin{array}{l}
\text{The sample space:} \\
\begin{align*}
S = \{&(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), \\
&(2,1), (2,2), (2,3), (2,4), (2,5), (2,6), \\
&(3,1), (3,2), (3,3), (3,4), (3,5), (3,6), \\
&(4,1), (4,2), (4,3), (4,4), (4,5), (4,6), \\
&(5,1), (5,2), (5,3), (5,4), (5,5), (5,6), \\
&(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)\}
\end{align*} \\\\
\text{The event spaces:} \\
E_{<5} = \{(1,1), (1,2), (1,3), (2,1), (2,2), (3,1)\} \\ E_{>8} = \{(3,6), (4,5), (4,6), (5,4), (5,5), (5,6), (6,3), (6,4), (6,5), (6,6)\} \\\\
\text{The event probability:} \\
P(E_{<5} \bigcup E_{>8}) = \frac{N(E_{<5})+N(E_{>8})}{N(S)} = \frac{16}{36} = \frac{4}{9}
\end{array}
\)

Answer

\(P(E_{<5} \bigcup E_{>8}) = \frac{4}{9}\)