Final answer:
The probability of less than 100 accidents occurring on this highway in a year is calculated using a normal approximation to the Poisson distribution due to the large expected number of accidents (156). The z-score for 99 accidents is computed and looked up in normal distribution tables.
Step-by-step explanation:
The question asks to determine the probability that less than 100 accidents will occur on a particular stretch of highway over the course of a year, given that the number of accidents follows a Poisson distribution with an average of 3 per week. Since 1 year is assumed to be 52 weeks, the expected number of accidents in one year is the weekly average (3) multiplied by 52, which equals 156 accidents per year on average.
To find the probability of having fewer than 100 accidents in a year, we might usually use the Poisson distribution formula directly.
However, because the average number of accidents is large (156), the calculation is complicated and we would typically use a normal approximation to the Poisson distribution when the mean is large to estimate this probability.
The normal approximation relies on the mean (μ = 156) and variance (σ^2 = 156, since the mean and variance are equal in a Poisson distribution), after which one would use these to find the z-score corresponding to 99 accidents and use standard normal distribution tables or technology to find the probability associated with this z-score.