a) What is the probability that at least 10 yellow jackets show up at a table in

Question

a) What is the probability that at least 10 yellow jackets show up at a table in the first 3 minutes?

b) What is the probability that the first yellow jacket shows up between one and two minutes from the start of a picnic?

c) What is the probability that the third yellow jacket shows up between one and two minutes from the start of a picnic? Express answer as a function of e.

There are 10 picnic tables at a park. Yellow jackets show up at each table at a constant rate of 2 per minutes. For the purposes of this problem, assume that all 10 tables always have people eating at them, and when a new gorup starts picnicking, there are no yellow jackets there at the beginning.

Please give random varibales & their distributions

Explanation / Answer

As we know, as per Poisson's distribution
m = lambda

P(X = x) = e^(-m)*m^x / x!

a)
lambda = 2 per minute
i.e 6 per 3 minutes

P(X = 10) = e^(-6) * 6^10 / 10! = 0.0413

b)
This means no yellow jacket showed up in first minute and at least one yellow jacket showed up in second minute

Probability = P(X = 0) * P(X > 0)
P(X = 0) = e^(-2) * 2^0 / 0! = 0.1353
P(X > 0) = 1 - 0.1353 = 0.8647

Probability = P(X = 0) * P(X > 0) = 0.1353 * 0.8647 = 0.117

c)
This means exactly 2 yellow jackets showed up in first minute and at least 1 yellow jacket showed up in second minute.

P(X = 2) = e^(-2) * 2^2 / 2! = 0.2707
P(X > 0) = 1 - 0.1353 = 0.8647

Required Probability = 0.2707 * 0.8647 = 0.2340

= (e^(-2) * 2^2 / 2!) * (1 - e^(-2) * 2^0 / 0!)
= 2*e^(-2) * (1 - e^(-2))
= 2*e^(-2) - 2*e^(-2)*e^(-2)

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