I need help with this question

Question

At a fair, Evie gets five darts she can toss at balloons. If there is a probability if 1/10 that any given dart she throws will hit a balloon, independent of her other throws, which of the following is the closest to the probability she will hit at least one balloon with her five darts?

1)41% 2)50% 3)54% 4)62%

Answer 1

Hi Carl!

Whenever you see an "at least" probability question, there is a very specific technique that you can use to make the problem easier: calculate the opposite probability!

It sounds strange, but here's how it works. We want to find the probability that she will hit at least one balloon. That means that usually we would have to calculate the probability of hitting exactly 1 balloon, then the probability of hitting exactly 2 balloons, then the probability of hitting exactly 3 balloons, and so on... too much work!

Instead, think about the opposite. The opposite outcome would be the probability that she hits 0 balloons. What's the chance of this happening? She has a 1/10 chance that any given dart will hit a balloon, so that means she has a 9/10 chance that any given dart will NOT hit a balloon. Since she has five darts, the only way she can hit 0 balloons, is if she misses every single shot! Since the events are independent, we can simply multiply the probabilities:

(9/10) * (9/10) * (9/10) * (9/10) * (9/10) = 59049/100000

Or approximately 59% chance that she will hit 0 balloons.

But that is the opposite of what we want, so we simply subtract the probability we found from 100% giving us:

The probability that she hits at least 1 balloon = 100% - the probability she hits 0 balloons = 100% - 59% = 41%

There we go: The probability that she will hit at least one balloon is 41%.

Hope that helps! Good luck!