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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.