Calculating the probability of an event

[DODGEVIPER13]

Homework Statement

A football team has a probability of .75 of winning when playing any of the other four teams in its conference. If the games are independent, what is the probability the team wins all the games?

Homework Equations

P(w1 u w2 u w3 u w4)=1-P(w1 u w2 u w3 u w4)' the ' mark stands for complement

The Attempt at a Solution

I found that the complement w=.25, and since it says independent that 1-p(w1)'p(w2)'p(w3)'p(w4)' so 1-.25^4=.996 probability of winning however the book says .75^4 why didn't they do as I did?

 

[DODGEVIPER13]

Like how do you know when to use .75^4 or what I did in problems what key words will tell me which one?

 

[vela]

You've calculated the probability that the team wins at least one game, not the probability that it wins all four games.

To win at least one game, it has to win game 1 OR game 2 OR game 3 OR game 4.
To win all four games, it has to win game 1 AND game 2 AND game 3 AND game 4.

 

[DODGEVIPER13]

Oh ok I get it the complement bar makes the union go to intersection so therefore I calculated the or and not the and.

 

[vela]

It really has nothing to do with taking the complement. You performed your calculation correctly; it just wasn't the right calculation to do. OR corresponds to the union while AND corresponds to the intersection, so $P(w1 \cup w2 \cup w3 \cup w4)$ is the probability of winning game 1 OR winning game 2 OR winning game 3 OR winning game 4.