Question

# The Titanic struck an iceberg and sank on its first voyage across the Atlantic in 1912. Some passengers got off the ship in lifeboats, but many died.

Two-way tables
The Titanic struck an iceberg and sank on its first voyage across the Atlantic in 1912. Some passengers got off the ship in lifeboats, but many died. The two-way table gives information about adult passengers who survived and who died, by class of travel.
$$\begin{array}{c|c} & First & Second & Third \\ \hline Yes & 197 & 94 & 151\\ \hline No & 122 & 167 & 476\\ \end{array}$$
Suppose we randomly select one of the adult passengers who rode on the Titanic. Given that the person selected was in first class, what's the probability that he or she survived?

2020-12-25
DEFINITIONS
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:$$\frac{197}{319}\approx0.6176=61.76\%$$