Example 4: using the probability rules

It is known that the probability that a child will buy a phantom comic is \(0.3\). If \(3\) children purchase comics at the same time, what is the probability that:

    a) all will purchase a phantom comic? 
    b) none will purchase a phantom comic? 
    c) only one child will purchase a phantom comic?

Solution:

Let \(A\) represent a child will purchase a Phantom comic. 

    a) \begin{eqnarray*} 
        P(\mbox{All purchase}) &=& P(A) \times P(A) \times P(A) \\ 
        &=& 0.3 \times 0.3 \times 0.3 \\ 
        &=& 0.027 
        \end{eqnarray*} 

    b) \begin{eqnarray*} 
        P(\mbox{none purchase}) &=& P(\bar{A})\times P(\bar{A})\times P(\bar{A}) \\ 
        &=& 0.7 \times 0.7 \times 0.7 \\ 
        &=& 0.343
        \end{eqnarray*} 

    c) \begin{eqnarray*}  
&& P(\mbox{1 purchase}) \\
&=& P(A\,,\ \bar{A}\,,\ \bar{A}) + P(\bar{A}\,,\ {A}\,,\ \bar{A}) + P(\bar{A}\,,\ \bar{A}\,,\ {A}) \\  
&=& 0.3\times 0.7 \times 0.7 + 0.7 \times 0.3\times 0.7 + 0.7\times0.7\times0.3 \\  
&=& 3 \times (0.3\times0.7\times0.7) \\  
&=& 0.441
\end{eqnarray*}