A Titanic disaster In $1912$the luxury liner Titanic, on its first voyage across the Atlantic, struck an iceberg and sank. Some passengers got off the ship in lifeboats, but many died. The two-way table gives information about adult passengers who lived and who died, by class of travel. Suppose we choose an adult passenger at random.

(a) Given that the person selected was in first class, what’s the probability that he or she survived?

(b) If the person selected survived, what’s the probability that he or she was a third-class passenger?

By class of travel, the two-way table provides information on adult passengers who lived and died. Let's say we pick a random adult passenger.

The number of favorable outcomes divided by the total number of possible outcomes equals probability. As a result, the following is a definition of condition probability:$P(A/B)=\frac{P(AandB)}{P\left(B\right)}$

The person is chosen first class, according to the question. Now we need to figure out how likely it is that "he or she survived."

Therefore,

$P(Firstclassandsurvive)=\frac{favorableoutcomes}{possibleoutcomes}=\frac{197}{1207}$

$P(Firstclass)=\frac{favorableoutcomes}{possibleoutcomes}=\frac{197+122}{1207}=\frac{319}{1207}$

As a result, the conditional probability is:

$$\begin{gathered} P(survived/Firstclass)=\frac{P(Firstclassandsurvived)}{P(Firstclass)} \\ P(survived/Firstclass)=\frac{{\displaystyle \frac{197}{1207}}}{{\displaystyle \frac{319}{1207}}} \\ P(survived/Firstclass)=\frac{197}{319} \\ P(survived/Firstclass)\approx 0.6167 \end{gathered}$$

As a result, the likelihood of the conclusion "he or she survived" is

$P(survived/Firstclass)=0.6167$

The person is chosen for survival based on the question. Now we need to figure out how likely it is that "he or she was a third-class traveler" was the case. Therefore,

$P(Thirdclassandsurvive)=\frac{favorableoutcomes}{possibleoutcomes}=\frac{151}{1207}$

$$\begin{gathered} P(Firstclass)=\frac{favorableoutcomes}{possibleoutcomes}=\frac{197+94+151}{1207} \\ P(Firstclass)=\frac{442}{1207} \end{gathered}$$

As a result, the conditional probability is:

$$\begin{gathered} P(Thirdclass/survived)=\frac{P(Thirdclassandsurvived)}{P(Firstclass)}=\frac{{\displaystyle \frac{151}{1207}}}{{\displaystyle \frac{442}{1207}}} \\ P(Thirdclass/survived)=\frac{151}{442}\approx 0.3416 \end{gathered}$$

As a result, the likelihood of the finding "he or she was a third-class traveler" is high.

$P(Thirdclass/survived)=0.3416$