AA be the probability of not wearing glasses and LaTeX: BB be the probability of having brown eyes. Suppose both events are independent such that LaTeX: P(A) = 0.90P ( A ) = 0.90 and LaTeX: P(B) = 0.70P ( B ) = 0.70.

LetLaTeX: AA be the probability of not wearing glasses and LaTeX: BB be the probability of having brown eyes. Suppose both events are independent such that LaTeX: P(A) = 0.90P ( A ) = 0.90  and LaTeX: P(B) = 0.70P ( B ) = 0.70. We want to find what proportion are not wearing glasses or have brown eyes (or both). This means we must find LaTeX: P(A cup B)P ( A ∪ B ).

In this case, we would use our union formula:

LaTeX: begin{eqnarray}

P(A cup B ) &=& P(A) + P(B) – P(Acap B) \

                           &=& 0.90+0.70-0.63 \

                          &=& 0.97

      end{eqnarray}                 P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) = 0.90 + 0.70 − 0.63 = 0.97

Hence, LaTeX: P(A cup B) = 0.97P ( A ∪ B ) = 0.97 (try it yourself before proceeding). Please create your own probability of the union example and explain why subtraction is needed in the union formula. In other words, why would it be incorrect to just simply add the probabilities outright? Relate the example to your major.