Question

according to a 2019 survey, the national basketball association (nba) is most popular among younger fans (aged 18 - 29) while major league baseball (mlb) and the national football league (nfl) are most popular for fans 30 years of age or older (statista.com). suppose that in a survey of 2,525 randomly selected adults in the united states, respondents are asked to provide their favorite professional sports league out of nba, mlb, and nfl. the results of the survey appear below.

favorite professional sports league
respondents age range nba mlb nfl
18 - 29 years old 235 125 245
30 - 54 years old 185 260 650
55+ 165 230 430
a. develop the joint probability table for these data. round your answers to three decimal places.

favorite professional sports league
respondents age range nba mlb nfl total
18 - 29 years old 0.093 0.050 0.097 0.240
30 - 54 years old 0.073 0.103 0.257 0.434
55+ 0.065 0.091 0.170 0.327
total 0.232 0.244 0.525

b. construct the marginal probabilities for each of the professional sports leagues (nba, mlb, and nfl). round your answers to three decimal places.

probability
nba 0.240
mlb 0.244
nfl 0.525

c. given that a respondent is 18 - 29 years of age, what is the probability that the respondents favorite professional sports league is mlb? round your answer to four decimal places.

d. given that a respondent is 55+ years of age, what is the probability that the respondents favorite professional sports league is mlb? round your answer to four decimal places.

e. given that the respondent states that mlb is their favorite professional sports league, what is the probability that the respondent is 18 - 29 years of age? round your answer to four decimal places.

f. given that the respondent states that the nfl is their favorite professional sports league, are they more likely to be aged 18 - 29, 30 - 54, or 55+? round your answers to four decimal places.

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$.

Step2: Solve part c

Let $A$ be the event that the favorite league is MLB and $B$ be the event that the age is 18 - 29 years old.
$P(A\cap B)$ is the joint - probability of being 18 - 29 and liking MLB, which from the joint - probability table is $0.050$.
$P(B)$ is the marginal probability of being 18 - 29, which is $0.240$.
So $P(A|B)=\frac{0.050}{0.240}\approx0.2083$.

Step3: Solve part d

Let $A$ be the event that the favorite league is MLB and $B$ be the event that the age is 55+ years old.
$P(A\cap B)$ is the joint - probability of being 55+ and liking MLB, which from the joint - probability table is $0.091$.
$P(B)$ is the marginal probability of being 55+, which is $0.327$.
So $P(A|B)=\frac{0.091}{0.327}\approx0.2783$.

Step4: Solve part e

Let $A$ be the event that the age is 18 - 29 years old and $B$ be the event that the favorite league is MLB.
$P(A\cap B)$ is the joint - probability of being 18 - 29 and liking MLB, which is $0.050$.
$P(B)$ is the marginal probability of liking MLB, which is $0.244$.
So $P(A|B)=\frac{0.050}{0.244}\approx0.2049$.

Step5: Solve part f

Let $A_1$ be the event that the age is 18 - 29, $A_2$ be the event that the age is 30 - 54, $A_3$ be the event that the age is 55+, and $B$ be the event that the favorite league is NFL.
$P(A_1|B)=\frac{P(A_1\cap B)}{P(B)}$, where $P(A_1\cap B) = 0.097$ and $P(B)=0.525$, so $P(A_1|B)=\frac{0.097}{0.525}\approx0.1848$.
$P(A_2|B)=\frac{P(A_2\cap B)}{P(B)}$, where $P(A_2\cap B)=0.257$ and $P(B) = 0.525$, so $P(A_2|B)=\frac{0.257}{0.525}\approx0.4895$.
$P(A_3|B)=\frac{P(A_3\cap B)}{P(B)}$, where $P(A_3\cap B)=0.170$ and $P(B)=0.525$, so $P(A_3|B)=\frac{0.170}{0.525}\approx0.3238$.
Since $0.4895>0.3238 > 0.1848$, they are more likely to be 30 - 54 years old.

Answer:

c. $0.2083$
d. $0.2783$
e. $0.2049$
f. 18 - 29: $0.1848$, 30 - 54: $0.4895$, 55+: $0.3238$, more likely to be 30 - 54 years old.