Chapter 5.3, Problem 66E

a.

To determine

To find:the probability that the result for “survived to second class” in the given table.

Answer to Problem 66E

  $P\left(\text{surived}|\text{Second class}\right)=0.3602$ .

Explanation of Solution

Given:

The two-way table gives information about adult passenger who lives and who died as in the given table:

Class of Travel	Survived	Died
First class	197	122
Second class	94	167
Third class	151	476

Concept used:

Probability is the number of favourable outcomes divided by the number of possible outcomes.

Thus, Definition condition probability:

  $P\left(A|B\right)=\frac{P\left(A\text{ and }B\right)}{P\left(B\right)}$

Calculation:

According to the question, the person is selected first class. Now to find the probability that the result for “survived to second class”.

Therefore,

  $P\left(\text{Second class and surive}\right)=\frac{\text{favourable outcomes}}{\text{possible outcomes}}=\frac{94}{1207}$

  $P\left(\text{First class}\right)=\frac{\text{favourable outcomes}}{\text{possible outcomes}}=\frac{94+167}{1207}=\frac{261}{1207}$

Thus, the conditional probability becomes:

  $\begin{array}{l}P\left(\text{surived}|\text{Second class}\right)=\frac{P\left(\text{Second class and survived}\right)}{P\left(\text{First class}\right)}\\ =\frac{94/1207}{261/1207}=\frac{94}{261}\approx 0.3602\end{array}$

Hence, the probability that the result for “survived to second class” is $P\left(\text{surived}|\text{Second class}\right)=0.3602$ .

b.

To determine

To find: the probability that the result for “survived” in the given table.

Answer to Problem 66E

  $P\left(\text{surived}\right)=0.3662$ .

Explanation of Solution

Given:

The two-way table gives information about adult passenger who lives and who died as in the given table:

Class of Travel	Survived	Died
First class	197	122
Second class	94	167
Third class	151	476

Calculation:

Probability is the number of favourable outcomes divided by the number of possible outcomes.

Thus, Definition condition probability:

  $P\left(A|B\right)=\frac{P\left(A\text{ and }B\right)}{P\left(B\right)}$

  $P\left(\text{surived}\right)=\frac{\text{favourable outcomes}}{\text{possible outcomes}}=\frac{442}{1207}\approx 0.3662$

Hence, the probability that the result for “survived” is $P\left(\text{surived}\right)=0.3662$ .

c.

To determine

To check: whether the events “a good chance” and “female” are independent.

Answer to Problem 66E

Yes, not independent.

Explanation of Solution

Given:

  $P\left(\text{surived}/\text{Second class}\right)=\frac{94}{261}\approx 0.3602$

  $P\left(\text{surived}\right)=\frac{\text{favourable outcomes}}{\text{possible outcomes}}=\frac{442}{1207}\approx 0.3662$

Calculation:

From part (a) and part (b),

  $P\left(\text{surived}/\text{Second class}\right)=\frac{94}{261}\approx 0.3602$

  $P\left(\text{surived}\right)=\frac{\text{favourable outcomes}}{\text{possible outcomes}}=\frac{442}{1207}\approx 0.3662$ .

There probability should be identical, thus they are not independent.