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The rate parameter, often denoted as λ (lambda), is a key concept in the Poisson distribution.
It represents the average number of occurrences of an event in a given interval of time or space.
In our exercise, the rate parameter λ is calculated by dividing the total number of car fatalities by the total billion miles traveled.
This gives us a rate of approximately 11.30 fatalities per billion miles.
This parameter is crucial as it is used in the Poisson probability formula to determine the likelihood of different numbers of events occurring.

The complement rule is a handy tool when calculating probabilities.
It states that the probability of an event occurring is one minus the probability of it not occurring.
In our context, to find the probability of at least one car fatality, we first calculate the probability of zero fatalities.
Then, we subtract that value from 1.
This simplifies our calculations because it's often easier to find the probability of zero occurrences than to calculate direct probabilities for multiple occurrences and add them up.

In the Poisson distribution, the probability of observing exactly k events is given by the formula: ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline \(P(X = k) = \frac{e^{-λ}λ^k}{k!}\)In our example, to find the probability of zero fatalities, we use k = 0.
Substituting λ = 11.30, we get \(P(X = 0) = e^{-11.30}\).
This results in a very small probability, approximately 1.24x10^{-5}.
Then, using the complement rule, we find \(P(X ≥ 1) = 1 - P(X = 0)\).
This yields a probability of approximately 0.999988, indicating that it is almost certain there will be at least one car fatality per billion miles traveled.

Interpreting statistical results is crucial for understanding the real-world implications.
In our problem, the final result indicates a probability of 0.999988 for at least one fatality in the next billion miles traveled.
This suggests that based on past data, it is almost guaranteed that there will be at least one fatality.
Such high probability underscores the importance of road safety and the need for measures to reduce fatalities.
It also shows how statistical tools like the Poisson distribution can provide valuable insights into real-world issues.