Poisson distribution and binomial distribution questions

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The discussion focuses on defining the Poisson and binomial probability distributions. The Poisson distribution is characterized by a mean (μ) that indicates the average number of events in a fixed interval, while the binomial distribution describes the probability of x successes in n independent trials with a success probability p. A specific example involves calculating the probability of finding no more than three defective nails from a sample of 400, given a defect rate of 0.15%. Additionally, the probability of exactly three lawn mowers being hired from a mean of 4.5 and the scenario where all six mowers are in use is explored. The thread emphasizes the importance of showing work for homework-related questions.
latkan
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Please help with this thanks :)

1.
(a) Define the Poisson probability distribution with mean μ.

(b) Write down the binomial distribution for x successes in n independent trials each with probability p of success.

(c) On average, 0.15% of the nails manufactured at a factory are known to be defective. If a random sample of 400 nails is inspected, what is the probability of there being no more than 3 defective nails?

4.
(a) Define the Poisson probability distribution with mean p.

(b) A tool hire shop has six lawn mowers which it hires out on a daily basis.The number of lawn mowers requested per day follows a Poisson probability distribution with mean 4.5. Find the probability that:

(i) exactly three lawn mowers are hired out on anyone day;
(ii) all lawn mowers are in use on anyone day.
 
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