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	60	194
2nd	93	24	117
3rd	81	57	138
total	308	141	449

a) what is the probability that a survivor was male?
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 male 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 male and in the 3rd class?
round final answer to 3 decimal places.
e) what is the probability that a survivor was a male or in the 3rd class?
round final answer to 3 decimal places.

Explanation:

Step1: Recall probability formula

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

Step2: Calculate probability for part a

The total number of survivors is $n = 449$, and the number of male survivors is $n_{male}=141$. So $P(\text{male})=\frac{141}{449}\approx0.314$

Step3: Calculate probability for part b

The number of 3rd - class survivors is $n_{3rd}=138$. So $P(\text{3rd class})=\frac{138}{449}\approx0.307$

Step4: Calculate probability for part c

We use the conditional - probability formula $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Here, $A$ is the event of being male and $B$ is the event of being in 3rd class. $n(A\cap B) = 57$ (number of male 3rd - class survivors) and $n(B)=138$ (number of 3rd - class survivors). So $P(\text{male}|\text{3rd class})=\frac{57}{138}\approx0.413$

Step5: Calculate probability for part d

The number of male 3rd - class survivors is $n = 57$. So $P(\text{male and 3rd class})=\frac{57}{449}\approx0.127$

Step6: Calculate probability for part e

We use the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. $P(A) = P(\text{male})=\frac{141}{449}$, $P(B)=P(\text{3rd class})=\frac{138}{449}$, and $P(A\cap B)=P(\text{male and 3rd class})=\frac{57}{449}$. Then $P(\text{male or 3rd class})=\frac{141 + 138-57}{449}=\frac{222}{449}\approx0.494$

Answer:

a) $0.314$
b) $0.307$
c) $0.413$
d) $0.127$
e) $0.494$