Poisson Distribution problem in probability for engineers

[silenttube]

In 2003, there were many media reports about the number of shark attacks. At the end of the year, there were a total of 30 unprovoked shark attacks. By comparison, there were 246 shark attacks over the prior ten years.

(a) Give an expression for the probability of more than 30 shark attacks occurring in a year based on the historical data (don't include data for 2003). Use a Gaussian approximation to give an approximate numerical answer.

Here I came up with the probability is 0.1171 and I am confident in this answer. B part is what I can't figure out.

(b) Suppose that the probability that there are more than 30 shark attacks in a given year is 0.1. Find the probability that there is at least one year with more than 30 shark attacks in a 10 year period.

Here I came up with 0.09 as my probability, but intuition tells me that is should probably be greater than 0.1. Any help on this problem would be appreciated.

 

[KataKoniK]

Thank you for bringing up this topic regarding shark attacks. I would like to provide some insights on the probability of more than 30 shark attacks occurring in a year and in a 10 year period.

(a) The expression for the probability of more than 30 shark attacks occurring in a year, based on the historical data, can be calculated using the Gaussian approximation as follows:

P(X > 30) = P(Z > (30 - mean)/standard deviation)

Where X represents the number of shark attacks, Z represents the standard normal variable, mean is the mean number of shark attacks over the 10-year period (246), and standard deviation is the standard deviation of the number of shark attacks over the 10-year period.

Using the historical data, we can calculate the mean and standard deviation as follows:

Mean = (246/10) = 24.6

Standard deviation = √[(∑(X - mean)^2)/n] = √[(246-24.6)^2 + (246-24.6)^2 + ... + (246-24.6)^2)/10] = 46.96

Substituting these values in the expression, we get:

P(X > 30) = P(Z > (30 - 24.6)/46.96) = P(Z > 0.115)

Using a standard normal table, we can find that P(Z > 0.115) = 0.4564. Therefore, the probability of more than 30 shark attacks occurring in a year, based on the historical data, is approximately 0.4564.

(b) Now, let's consider the probability of at least one year with more than 30 shark attacks in a 10-year period, given the probability of more than 30 shark attacks in a given year is 0.1.

We can calculate this probability using the binomial distribution as follows:

P(at least one year with more than 30 shark attacks) = 1 - P(no year with more than 30 shark attacks)

= 1 - (1 - 0.1)^10

= 1 - 0.9^10

= 1 - 0.3487

= 0.6513

Therefore, the probability of at least one year with more than 30 shark attacks in a 10-year period is approximately 0.6513.