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# Probability

### Example 2: using probability rules

The probability that a certain man will live $$10$$ more years is $$0.25$$ and the probability that his wife will live $$10$$ more years is $$0.3$$. What is the probability that:
a) both will be alive in $$10$$ years time?
b) only the wife will be alive in $$10$$ years?
c) neither will be alive in $$10$$ years?
d) what assumptions do you need to make for these questions?

Solution:

This question uses the multiplication rule. The use of a tree diagram will assist.
Let $$A$$ represent the man lives for $$10$$ or more years, and
let $$B$$ represent the woman lives for $$10$$ or more years. a) \begin{eqnarray*}
P(A \mbox{ and } B)
&=& P(A)\times P(B) \\
&=& 0.25 \times 0.3 \\
&=& 0.075
\end{eqnarray*}

b) \begin{eqnarray*}
P(\bar{A} \mbox{ and } B)
&=& P(\bar{A}) \times P(B) \\
&=& 0.75 \times 0.3 \\
&=& 0.225
\end{eqnarray*}

c)  \begin{eqnarray*}
P(\bar{A} \mbox{ and } \bar{B})
&=& P(\bar{A}) \times P(\bar{B}) \\
&=& 0.75 \times 0.7 \\
&=& 0.525
\end{eqnarray*}

d) The assumption is that the life span of both people are independent.