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

Answer:

1. The probability that Isaac will not become a Hurricane by the time it reaches the Gulf of Mexico is 0,31

2. The probability of the storm becoming a hurricane is reduced if it passes over a landmass.

Step-by-step explanation:

Hello!

1. In the first part of the problem you have an event A: "The tropical storms Isaac will become a Hurricane by the time it reaches the Gulf of Mexico" and it's associated probability is P(A): 0,69

The event "The tropical storm Isaac will not become a hurricane by the time it reaches the Gulf of Mexico" is the complement of A, symbolized as $A^{c}$. What we need to calculate now is the associated probability of the complement

Applying Axiomatic probability it follows that:

$P(A) + P(A^{c}) = 1$

From the formula we can clear the desired probability:

$P(A^{c}) = 1 - P(A)$

$P(A^{c}) = 1 - 0.69$

$P(A^{c}) = 0,31$

The probability that Isaac will not become a hurricane by the time it reaches the Gulf of Mexico is 0,31

2. The second part asks: How did passing over Cuba alter the probability that Isaac would become a hurricane by the time it reached the Gulf of Mexico?

Information:

"Isaac will pass directly over Cuba before reaching the Gulf of Mexico"

"Hurricanes that reach the Gulf of Mexico have a 0,08 probability of having passed over Cuba."

"Tropical storms that reach the Gulf of Mexico have a 0,20 probability of having passed over Cuba"

Now we have three events to take into consideration

$A$ "The tropical storms Isaac will become a Hurricane by the time it reaches the Gulf of Mexico"

$A^{c}$ "The tropical storm Isaac will not become a hurricane by the time it reaches the Gulf of Mexico"

$B$ "The tropical storm has passed over Cuba"

And the probabilities

$P(A) = 0,69$

$P(A^{c}) = 0,31$

         >The events A and B are dependable, because "having passed over Cuba" affects the probability of "The tropical storm becoming an Hurricane", the same happens between $A^{c}$ and B. Therefore the later probabilities are conditional probabilities and we can symbolize the as:

$P(B/A) = 0,08$

$P(B/A^{c}) = 0,20$

The probability we need to calculate for this part is symbolized as:

P(B∩A) "The tropical storm has passed over Cuba" and "The tropical storms Isaac will become a Hurricane by the time it reaches the Gulf of Mexico"

The conditional probability in defined as:

P(B/A) = P(B ∩ A)/P(A)

Then:

P(B∩A) = P(B/A)*P(A)

P(B∩A) = 0,08*0,69

P(B∩A) = 0,0552

The probability of the storm becoming a hurracane is reduced if it passes over a landmass.

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