4. Probability statements

In order to make probability statements we need a probability model for the process of interest. Constructing these models is not material for this course! But utilizing them is.

We will start with a model for annual hurricane frequency.

 

Table 1: Frequencies of yearly hurricane totals for the North Atlantic, 1935-1995

Number of storms	4	5	6	7	8	9	10	11	12	13	14	16	18	19
Frequency	1	2	7	6	9	6	6	10	5	4	2	1	1	1

This table indicates that the were 5 years with 12 storms etc.

Question: what's the probability there are 17 hurricanes next year?

We haven't had a year with exactly 17 hurricanes. We have had 16, and we have had 18 so 17 must be possible. Estimating it as 0/51 is a bit naive.

We will smooth these observations and make a probability distribution/model for the number of hurricanes.

Here is a probability model:

 

Table 2: Probability model for yearly hurricane frequencies for the North Atlantic

Number of storms	0	1	2	3	4	5	6	7	8	9	10
Probability	0.000	0.001	0.003	0.010	0.024	0.046	0.073	0.100	0.121	0.129	0.124
Number of storms	11	12	13	14	15	16	17	18	19	20
	0.109	0.087	0.065	0.044	0.028	0.017	0.010	0.005	0.003	0.001

Note that probabilities lie between 0 and 1 and sum to 1.

Use it to calculate probabilities of events. Make an assumption: the model can describe the probability of events next year.

What's the probability of less than 6 hurricanes next year?

What's the probability of between 8 and 12 hurricanes next year?

Define a disaster year as one with more than 14 hurricanes. What's the probability that we don't have a disaster year?

If we call the number of hurricanes next year X, then these questions can be written as

Find P(X < 6).

Find $P(8 \le X \le 12)$.

Find $P(X < 15) = 1 - P(X \ge 15)$.

Method: identify the components that correspond to the event of interest and sum the probabilities.

We find probabilities of complicated events by splitting them down into simple events and then adding the probabilities of those simple events.

Other questions, like what is the average number of hurricanes in a year, we will do in Class 10.