SOLUTION: The probability that a certain region in Mexico will be hit by a hurricane in any given year is 0.07. What is the probability that the region will be hit by at least one hurricane

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Question 341421: The probability that a certain region in Mexico will be hit by a hurricane in any given year is 0.07. What is the probability that the region will be hit by at least one hurricane in the next 14 years?
Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

The probability of successes in trials where is the probability of success on any given trial is given by:

Where is the number of combinations of things taken at a time and is calculated by

Here you want to consider the probability over 14 trials where the probability of success (yes, in this context getting hit by a hurricane would be considered a 'success') is 0.07.

The calculation outlined above gives the probability for a specific number of successes. Where you have situation such as in this problem where you are asked for the probability of "at least" some number, then you have to consider the probability of that number, plus the probability of one more, plus the probability of one more than that...up to the probability of success on every trial.

Symbolically, here is what you would need:

I'm sure you will agree that is uglier than a mud fence, and you may take it on faith that calculating such a thing has all of the entertainment value of watching paint dry. But fear not! There is a simpler way.

The only possibility other than getting hit by at least one is to get hit by none at all! That means the probability of getting hit by at least one PLUS the probability of getting hit by none at all is equal to 1. In other words:

And here is the 'so what':

And since and , the above reduces to:

I'll leave you alone to spend a little quality time with your calculator.

John

My calculator said it, I believe it, that settles it