Tires Probability Question

[Koshilia]

Question:
You recently bought a new set of four tires from a manufacturer who just announced a recall because 2% of that particular brand of tires are defective. What is the probability that at least one of your tires is defective? You may assume that the tires are defective independently of one another.

Since a tire cannot have one tire defective and two tires defective at the same time, I considered the events to be disjoint. I used the addition rule:
P(1 tire is defective) + P(2 tires are defective) + P(3 tires are defective) + P(4 tires are defective)

To find the probability of two tires being defective, I multiplied .02*.02. To find the probability that three tires being defective, I multiplied .02*.02*.02. And, so on for the four tires.. This is because the problem stated that they are independent.

I ended up with the wrong answer. Can someone please help me?

 

[Dason]

.02*.02 gives you the probability that the first two tires you look at are both defective. You haven't specified what happened with the other two tires - they could be good it they could be bad. But if you're calculating the probability that exactly two are bad then you can't allow the other two to be bad. You also don't Account for how many ways there could be two bad tires. It could be the front two. It could be the back two... And so on. Hopefully this helps you understand some of the shortcomings of your answer