A Titanic disaster Refer to Exercise 64.

(a) Find P(survived | second class).

(b) Find P(survived).

(c) Use your answers to (a) and (b) to determine whether the events “survived” and “second class” are independent. Explain your reasoning.

The $595$kids who answered to a school survey regarding breakfast eating are shown in the two-way table below. Assume we choose a student at random. Consider the following events: $B$: eats breakfast on a regular basis, and $M$: is a man.

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: $P\left(A\right|B)=\frac{P(AandB)}{P\left(B\right)}$

The person is chosen first class, according to the question. Now we must determine the likelihood that the outcome for "survived to second class" will be positive. Therefore,

$P(Firstclass)=\frac{favourableoutcomes}{possibleoutcomes}=\frac{94+167}{1207}=\frac{261}{1207}$

As a result, the conditional probability is:

$$\begin{gathered} P\left(surived\right|Secondclass)=\frac{P(Secondclassandsurvived)}{P(Firstclass)} \\ P(surived|Secondclass)=\frac{{\displaystyle \frac{94}{1207}}}{{\displaystyle \frac{261}{1207}}}=\frac{94}{261}\approx 03602 \end{gathered}$$

As a result, there's a good chance that the result for "survived to second class" will be positive.

$P\left(surived\right|Secondclass)=0.3602$

$P\left(survived\right)=\frac{favorableoutcomes}{possibleoutcomes}=\frac{442}{1207}\approx 0.3662$

Part (a) and part (b),

$P\left(\frac{survived}{Secondclass}\right)=\frac{94}{261}\approx 0.3602$

$P\left(surived\right)=\frac{favourableoutcomes}{possibleoutcomes}=\frac{442}{1207}\approx 0.3660$

They should have the same probability, hence they are not independent.