Final answer:
In another unrelated scenario, events involving drawing cards and coin tosses show that picking a blue card followed by landing a head is mutually exclusive from picking a red or green card followed by landing a head, but is not mutually exclusive from picking a red or blue card followed by landing a head.
Step-by-step explanation:
The question you've asked seems to encompass elements of sport rules and probability, hence it might be slightly confusing.
But, to clarify, you seem to be referring to two separate scenarios: one is a game situation involving a blue card, and the other involves a probability problem with fans wearing colors and rooting for teams. Let's focus on the probability aspect which is a mathematical question.
To decide whether the events of fans rooting for the away team (A) and wearing blue (B) are independent, we need to look at the definition of independence in probability. Events A and B are independent if the probability of one event occurring does not affect the probability of the other.
We have that 20 percent of the fans are wearing blue and rooting for the away team (given by the intersection of A and B), and 67 percent of the fans rooting for the away team are wearing blue (which is within event A only).
To test independence, we can calculate P(A) * P(B) and compare it to P(A ∩ B). If these are not equal, the events are not independent. With regard to mutual exclusivity, events A and B are not mutually exclusive because there are fans that are both wearing blue and rooting for the away team (intersection of A and B is not empty).
When we switch to the second probability scenario involving a deck of cards and a coin, if we let A be the event of picking a blue card and flipping a coin and it lands on heads, P(A) can be found by multiplying the probability of drawing a blue card by the probability of flipping a head.
Events A and B (picking a red or green card and landing a head on the coin toss) are mutually exclusive because they cannot happen at the same time; the card cannot be both blue and red/green.
However, A and C (picking a red or blue card and landing a head on the coin toss) are not mutually exclusive because the blue card is part of both A and C.