$P = \dfrac{\text{number of favorable ways}}{\text{total number of ways}}$

Poisson Probability Distribution

The number of occurrences in a given time interval or in a given space can be modeled using Poisson Distribution if the following conditions are being satisfied:

  • The events occur at random.
  • The events are independent from one another.
  • The average rate of occurrences is constant.
  • There are no simultaneous occurrences.

 

The Poisson distribution is defined as

$P(x) = \dfrac{e^{-\mu} \mu^x}{x!}$

where x is a discrete random variable

P(x) = probability for x occurrences
μ = the mean number of occurrences

Problem
In a common carnival game, a player tosses a penny from a distance of about 5 feet onto the surface of a table ruled in 1-inch squares. If the penny (3/4 inch in diameter) falls entirely inside a square, the player receives 5 cents but does not get his penny back; otherwise he loses his penny. If the penny lands on the table, what is his chance to win?

A.   5/16 C.   9/256
B.   1/16 D.   3/128

 

Problem
From where he stands, one step toward the cliff would send the drunken man over the edge. He takes random steps, either toward or away from the cliff. At any step his probability of taking a step away is 2/3, of a step toward the cliff 1/3. What is his chance of escaping the cliff?

A.   2/27 C.   4/27
B.   107/243 D.   1/2

 

Problem
Samuel Pepys wrote Isaac Newton to ask which of three events is more likely: that a person get (a) at least 1 six when 6 dice are rolled (b) at least two sixes when 12 dice are rolled, or (c) at least 3 sixes when 18 dice are rolled. What is the answer?

A.   (a) is more likely than (b) and (c)
B.   (b) is more likely than (a) and (c)
C.   (c) is more likely than (a) and (b)
D.   (a), (b), and (c) are equally likely