Question # In 1912 the Titanic struck an iceberg and sank on its first voyage. Some passengers got off the ship in lifeboats, but many died. The following two-wa

Two-way tables
ANSWERED In 1912 the Titanic struck an iceberg and sank on its first voyage. Some passengers got off the ship in lifeboats, but many died. The following two-way table gives information about adult passengers who survived and who died, by class of travel.
$$\begin{array} {lc} & \text{Class} \ \text {Survived } & \begin{array}{c|c|c|c} & \text { First } & \text { Second } & \text { Third } \\ \hline \text { Yes } & 197 & 94 & 151 \\ \hline \text { No } & 122 & 167 & 476 \end{array}\ \end{array}$$
Suppose we randomly select one of the adult passengers who rode on the Titanic. Define event D as getting a person who died and event F as getting a passenger in first class. Find P(not a passenger in first class or survived). 2021-02-13
DEFINITIONS
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.
$$P(F^C\ and\ D^c)=\frac{\text{# of favorable outcomes}}{\text{# of possible outcomes}}=\frac{245}{1207}$$
$$P(F^C\ and\ D^c)=P(F^c)+P(D^c)-P(F^c\ and\ D^c)\\ =\frac{888}{1207}+\frac{442}{1207}-\frac{245}{1207}\\ =\frac{888+442-245}{1207}\\=\frac{1085}{1207}\\\approx0.8989\\=89.89\%$$
Result:
$$\frac{1085}{1207}\approx0.8989=89.89\%$$ 2021-08-12

Given:

$$\begin{array}{|c|c|}\hline & \text{First} & \text{Second} & \text{Third} & \text{Total} \\ \hline \text{Survived} & 197 & 94 & 151 & 442 \\ \hline \text{Not survived} & 122 & 167 & 476 & 765 \\ \hline \text{Total} & 319 & 261 & 627 & 1207 \\ \hline \end{array}$$
$$94+151=245$$ of the 1207 people on the Titanic were not a passenger in first class and survived.
The probability is the number of favorable outromes divided by the number of possible outromes:
$$\displaystyle{P}{\left({F}^{{C}}{\quad\text{and}\quad}{D}^{{c}}\right)}=\#$$ of favorable outromes/# of possible outromes$$\displaystyle=\frac{{245}}{{1207}}\sim-{0.2030}={20.30}\%$$