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# Probability

## 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*}