Question

In a basket ball arena

• 70% of the fans are rooting for the home team.

• 25% of the fans are wearing blue.

• 20% of the fans are wearing blue and are rooting for the away team.

• Of the fans rooting for the away team, 67% are wearing blue.

Let A be the event that a fan is rooting for the away team.

Let B be the event that a fan is wearing blue.

Are the events of rooting for the away team and wearing blue independent? Are they mutually exclusive?

Short answer

Events of rooting away for away team & wearing blue are neither independent, nor mutually exclusive.

Step-by-step solution

Basics 

Mutually exclusive Events are those which cannot occur simultaneously, like getting both a head & a tail at single toss of a fair die

Independent Events are those, in which occurrence or non occurrence of an event doesn't effect the occurrence or probability dynamics of the other event. Eg : Probability of happy mood of two unrelated people.

Explanation 

Event A - Hooting for away team, & event B - wearing blue : are not mutually exclusive, as they can occur at the same time. Since of the fans hooting for away team, 67% are wearing blue, so there are people who are hooting for away team & also wearing blue.

Event A - Hooting for away team, & event B - wearing blue : are not independent, as event A effects the occurrence of event B, since people wearing blue are 67% among those hooting for away team itself.