Question

the carolina panthers football team won 41% of their football games in the 2022-2023 season. during that time, they won 5 of the 9 games they played at home. are the events \panthers win the game\ and \panthers win the game when playing at home\ dependent or independent events? select the best answer and explanation below.

a) the events are dependent because the panthers won 56% of games when they played at home and 41% of the games overall.

b) the events are independent because the panthers won 56% of games when they played at home and 41% of the games overall.

c) the events are dependent because the panthers won 41% of games when they played at home and 41% of the games overall.

d) the events are independent because the panthers won 41% of games when they played at home and 41% of the games overall.

Explanation:

To determine if two events \( A \) (Panthers win the game) and \( B \) (Panthers win the game when playing at home) are independent, we check if \( P(A|B)=P(A) \).

First, calculate the percentage of home wins: They won 5 out of 9 home games, so \( \frac{5}{9}\approx 56\% \). The overall win percentage is 41%.

For independent events, the conditional probability \( P(A|B) \) should equal the marginal probability \( P(A) \). Here, \( 56\%
eq 41\% \), so the probability of winning changes when we know it's a home game. Thus, the events are dependent because the home win percentage (56%) is different from the overall win percentage (41%).

Answer:

a) The events are dependent because the Panthers won 56% of games when they played at home and 41% of the games overall.