Complement rule:

\(P(A^c)=P(not\ A)=1-P(A)\)

Definition conditional probability:

\(P(B|A)=\frac{P(A\cap B}{P(A)}=\frac{P(A\ and\ B)}{P(A)}\)

SOLUTION

Given:

\(\begin{array}{l|ccc|c} & \text { First } & \text { Second } & \text { Third } & \text{Total} \\ \hline \text {Survived} & 197 & 94 & 151 & 442 \\ \text {Not survived } & 122 & 167 & 476 & 765 \\ \hline \text {Total} & 319 & 261 & 627 & 1207 \end{array}\)

319 of the 1207 people on the Titanic were a passenger in first class.

The probabillity is the number of favorable outcomes divided by the number of possible outcomes:

\(P(F)=\frac{\text{# of favorable outcomes}}{\text{# of possible outcomes}}=\frac{319}{1207}\)

Complement rule:

\(P(not \ A)=1-P(A)\)

Using the complement rule, we then obtain: \(P(F^C)=1-P(F)=1-\frac{319}{1207}=\frac{888}{1207}\)

442 of the 1207 people on the Titanic survived.

\(P(D^C)=\frac{\text{# of favorable outcomes}}{\text{# of possible outcomes}}=\frac{442}{1207}\)

94+151=245 of the 1207 people on the Titanic were not a passenger in first class and survived.

\(P(F^C\ and\ D^c)=\frac{\text{# of favorable outcomes}}{\text{# of possible outcomes}}=\frac{245}{1207}\)

Use the general addition rule:

\(P(F^C\ and\ D^c)=P(F^c)+P(D^c)-P(F^c\ and\ D^c)\\ =\frac{888}{1207}+\frac{442}{1207}-\frac{245}{1207}\\ =\frac{888+442-245}{1207}\\=\frac{1085}{1207}\\\approx0.8989\\=89.89\%\)

Result:

\(\frac{1085}{1207}\approx0.8989=89.89\%\)