Final answer:
The question involves calculating probabilities using the Poisson distribution, requiring the use of a distribution table or statistical software, and rounding results to four decimal places. It also notes scenarios where the Poisson may approximate the binomial distribution.
Step-by-step explanation:
The question is related to using the Poisson distribution to calculate probabilities for a given statistical scenario. In finding probabilities, such as P(x = 3), P(1 < x < 4), and P(x ≥ 8), one must reference a Poisson distribution table or use a statistical function like poissoncdf. These calculations are important in scenarios where the number of events within a specific interval of time or space is being measured, given a known average rate, and assuming each event occurs independently from the last.
To calculate P(x = 3), you would look up the value corresponding to 3 in the Poisson table for your specific λ (average rate) value. To find P(1 < x < 4), you would calculate the cumulative probability up to 3 and subtract the cumulative probability up to 1. For P(x ≥ 8), you would either look for the cumulative probability of 7 and subtract it from 1, or use a statistical software function. Remember to round your answers to four decimal places, which is a common convention for presenting probabilities in statistics.
It's also noted that in some situations, the Poisson distribution can be used to approximate the binomial distribution when certain conditions are met, specifically when the number of trials (n) is large and the probability of success in each trial (p) is small.