Question

In August 2012, tropical storm Isaac formed in the Caribbean and was headed for the Gulf of Mexico. There was an initial probability of .69 that Isaac would become a hurricane by the time it reached the Gulf of Mexico (National Hurricane Center website, August 21, 2012).

What was the probability that Isaac would not become a hurricane but remain a tropical storm when it reached the Gulf of Mexico?

Two days later, the National Hurricane Center projected the path of Isaac would pass directly over Cuba before reaching the Gulf of Mexico. How did passing over Cuba alter the probability that Isaac would become a hurricane by the time it reached the Gulf of Mexico? Use the following probabilities to answer this question. Hurricanes that reach the Gulf of Mexico have a .08 probability of having passed over Cuba. Tropical storms that reach the Gulf of Mexico have a .20 probability of having passed over Cuba.

What happens to the probability of becoming a hurricane when a tropical storm passes over a landmass such as Cuba?

  

Answer 1

Concepts and reason

The probability of an event A represents the likelihood that a random experiment will result in an outcome in the set A relative to the sample space of the random experiment, if we have prior information that the outcome of the random experiment must be in a set B of S, then this information must be used to re-appraise the likelihood that the outcome will also be in B.

This re-appraised probability is denoted by and is read as the conditional probability of the event A, given that the event B has already happened.

Fundamentals

Multiplication theorem of probability:

For two events A and B,

Here, represents conditional probability of occurrence of B when the event has already happened and is the conditional probability of happening of A, given that B has already happened.

Bayes’ theorem tells about the probability on an event based on prior knowledge with some conditions that might be related to that event.

If are mutually disjoint events with, then for any arbitrary event A which is a subset of such that á

Hence,

Compute the probability that Isaac would not become hurricane.

Consider that:

It is given in the problem that,

and

and

So, the required probability by using Baye’s Theorem is given as below:

Compute the probability of becoming a hurricane when a tropical storm passes over a landmass such as Cuba.

$P(4 B)- P(4)P(B| A) P(B) P(A\B) 0.69 x 0.08 0.1172 0.4710 $

Probability that Isaac would become a hurricane but remain a tropical storm when it reached the Gulf of Mexico is 0.69.

Therefore, the probability of becoming a hurricane when a tropical storm passes over a landmass such as Cuba decreased from 0.69 to 0.4710.

Ans:

Probability that Isaac would not become a hurricane but remain a tropical storm when it reached the Gulf of Mexico is 0.31.

Probability that Isaac would become a hurricane by the time it reached the Gulf of Mexico is 0.1172.

Probability of becoming a hurricane when a tropical storm passes over a landmass such as Cuba is 0.4710.

Probability of becoming a hurricane when a tropical storm passes over a landmass such as Cuba decreased from 0.69 to 0.4710