Definition conditional probability:

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

SOLUTION

\(\begin{array}{c|ccc|c} & First & Second & Third &Total\\ \hline Yes & 197 & 94 & 151&442\\ No & 122 & 167 & 476&765\\ \hline Total & 319 & 261 & 627 & 1207 \end{array}\)

F=First class

S=Survived

We note that the table contains information about 1207 passengers (given in the bottom right corner of the table).

Moreover, 319 of the 1207 passengers are first class passengers, because 319 is mentioned in the row *Total” and in the column *First” of the table. The probability is the number of favorable outcomes divided by the number of possible outcomes:

\(P(F)=\frac{\# \text{ of favorable outcomes}}{\# \text{ of possible outcomes}}=\frac{319}{1207}\) Next, we note that 197 of the 1207 passengers are first class passengers that survived, because 197 is mentioned in the row ”Yes” and in the column ” First” of the given table.

\(P(F \text{ and } S)=\frac{\# \text{ of favorable outcomes}}{\# \text{ of possible outcomes}}=\frac{197}{1207}\)

Use the definition of conditional probability:

\(P(S|F)=\frac{P(F \text{ and } S)}{P(F)}=\frac{\frac{197}{1207}}{\frac{319}{1207}}=\frac{197}{319}\approx0.6176=61.76\%\)

Answer:\(\frac{197}{319}\approx0.6176=61.76\%\)