Question

The Poisson Distribution

The Poisson Distribution is a Discrete Probability Distribution that is commonly applied when a series of trials/experiments occur over an interval. (The number of meteors per hour; hailstones per acre). Here, the average time/distance/etc. between each event must be known.
This type of distribution may be used if the following conditions apply:
Each event is independent.
The average rate is constant (events per interval).
Two events cannot occur at the same time. . .
Apply the Poisson Distribution to a scenario.
On average, a baking student accidentally drops three pieces of egg shell into the batter of every two cakes made. .
The conditions of a Poisson Distribution are met:
• The baking of a cake does not affect the outcome of any other cake. (independent events)
3 The average rate is constant: 3 pieces per 2 cakes rate = = 1.5 2
Baking (and dropping eggshells) is a sequential process. (Two events cannot occur at the same time) ) .
Use the scenario above to determine the expected value (u) and selected probabilities below. You may wish to use the Poisson Distribution Calculator hosted by the University of lowa's Department of Mathematical Sciences. Remember: the formatting of this calculator may vary slightly from what is used in class. (link: Poisson Distribution Calculator 2)
a. On average, how many egg shells do you expect to be in a single cake? = (decimal answers only)
b. If a single cake is bought, what is the probability of finding 0 egg shell pieces in it? P(X = 0) = (include four decimal places) =
c. If a single cake is bought, what is the probability of finding more than two egg shell pieces in it? P(X > 2) = (include four decimal places)

Answer 1

The Poisson distribution is a probability distribution that models the number of events occurring in a fixed interval of time or space, given a known average rate and independence of events. In this scenario, the number of eggshell pieces in a cake can be modeled using the Poisson distribution with an average rate of 1.5 eggshell pieces per cake. The probabilities of finding 0 eggshell pieces and more than two eggshell pieces can be calculated using the Poisson distribution formula.

Step-by-step explanation:

The Poisson distribution is a probability distribution that models the number of events occurring in a fixed interval of time or space, given a known average rate and independence of events. In this scenario, the average rate is 3 eggshell pieces per 2 cakes made. To find the expected value, or average number of eggshell pieces in a single cake, we can calculate λ, the average rate per one cake, by dividing 3 by 2. So, λ = 3/2 = 1.5 eggshell pieces per cake.

To find the probability of finding 0 eggshell pieces in a single cake, we can use the Poisson distribution formula:

P(X = 0) = (e^(-λ) * λ^0) / 0! = e^(-1.5)

To find the probability of finding more than two eggshell pieces in a single cake, we can use the complementary probability:

P(X > 2) = 1 - P(X ≤ 2)

We can use a Poisson distribution calculator or tables to find the exact values for these probabilities.