Question

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.

Short answer

Part (a) P (survived | Second class) =$0.3602$

Part (b) P (survived)=$0.3662$

Part (c) Yes, not independent.

Step-by-step solution

Part (a) Step 1. Given Information

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.

Part (a) Step 2. Concept Used

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

Part (a) Step 3. Calculation

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,

$P(Secondclassandsurvive)=\frac{favorableoutcomes}{possibleoutcomes}=\frac{94}{1207}$

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

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.

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

Part (b) Step 1. Calculation

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

Part (c) Step 1. Calculation

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.