Question

the number of people who survived the titanic based on class and gender is in the following table. suppose a person is picked at random from the survivors.

class	female	male	total
1st	134	58	192
2nd	94	26	120
3rd	79	58	137
total	307	142	449

a) what is the probability that a survivor was female?
round final answer to 3 decimal places.
b) what is the probability that a survivor was in the 3rd class?
round final answer to 3 decimal places.
c) what is the probability that a survivor was a female given that the person was in 3rd class?
round final answer to 3 decimal places.
d) what is the probability that a survivor was a female and in the 3rd class?
round final answer to 3 decimal places.
e) what is the probability that a survivor was a female or in the 3rd class?
round final answer to 3 decimal places.
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Explanation:

Step1: Define probability formula

Probability $P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

Step2: Calculate probability of female survivor (a)

The number of female survivors is 307 and total survivors is 449. So $P(\text{Female})=\frac{307}{449}\approx0.684$

Step3: Calculate probability of 3rd - class survivor (b)

The number of 3rd - class survivors is 137 and total survivors is 449. So $P(\text{3rd Class})=\frac{137}{449}\approx0.305$

Step4: Calculate conditional probability (c)

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Here, $A$ is being female and $B$ is being in 3rd class. The number of female 3rd - class survivors is 79 and number of 3rd - class survivors is 137. So $P(\text{Female}|\text{3rd Class})=\frac{79}{137}\approx0.577$

Step5: Calculate probability of female and 3rd - class (d)

The number of female 3rd - class survivors is 79 and total survivors is 449. So $P(\text{Female}\cap\text{3rd Class})=\frac{79}{449}\approx0.176$

Step6: Calculate probability of female or 3rd - class (e)

Use the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Here, $P(A) = P(\text{Female})=\frac{307}{449}$, $P(B)=P(\text{3rd Class})=\frac{137}{449}$ and $P(A\cap B)=P(\text{Female}\cap\text{3rd Class})=\frac{79}{449}$. Then $P(\text{Female}\cup\text{3rd Class})=\frac{307 + 137-79}{449}=\frac{365}{449}\approx0.813$

Answer:

a) 0.684
b) 0.305
c) 0.577
d) 0.176
e) 0.813