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The concept of probability, its meaning, interpretation, and measurement through simulations. It covers objective and subjective interpretations, simulating probabilities, and finding probabilities using simulations. Two practical examples are provided to illustrate the concepts. Useful for students in statistics, mathematics, or data science courses.
Typology: Study notes
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We now consider the problem of making inference. That is, using sample information to infer about the population with a certain measure of reliability.
Formulate Theories
Collect Data
Summarize Results
Interpret Results/Make Decisions
Objective Interpretation: The probability value of an event is equated with the relative frequency of occurrence of the event in the long run under constant causal conditions. The objective interpretation of probability applies only to repeatable events and not to unique events. Subjective Interpretation: The subjective interpretation of probability (also called the personal interpretation) relates probability to degree of personal belief.
Steps for finding probabilities using simulation:
1. Specify a model for the individual outcomes of the underlying random phenomenon. 2. Outline how to simulate an individual outcome and how to represent a single repetition of the random process. 3. Simulate many repetitions and estimate the probability of an event by its relative frequency.
Let’s do it! 8.
A couple plans to have children. They would like to have a boy to be able to pass on the family name. After some discussion, they decide to continue to have children until they have a boy or until they have three children, whichever comes first. What is the probability that they will have a boy among their children?
1. Specify a model for the individual outcomes. 2. Simulate individual outcomes.
Let’s do it! 8.
There are three doors. Behind one door is a car. Behind each of the other two doors is a goat. As a contestant, you are asked to select a door, with the idea that you will receive the prize that is behind that door. The game host knows what is behind each door. After you select a door, the host opens one of the remaining doors that has a goat behind it. Note that no matter which door you select, at least one of the remaining doors has a goat behind it for the host to open. The host then gives you the following two options:
1. Stay with the door you originally selected and receive the prize behind it. 2. Switch to the other remaining closed door and receive the prize behind it.
What is the probability of winning the car if you stay? What is the probability of winning the car if you switch? Will switching increase your chance of winning the car?
Random Variables
Definition A random variable is a rule that assigns one (and only one) numerical value to each simple event of an experiment
Example Consider the experiment: Toss two coins Let X be the random variable denoting the number of tails
Simple Event Numerical Value
Definition The probability distribution of a discrete random variable is a graph, table, or formula that specifies the probability associated with each possible value the random variable can take. Example Let X be the number of people in a household for a certain community. Consider the following probability distribution for X, which assumes that there are no more than seven people in a household. X = x 1 2 3 4 5 6 7 PX = x 0. 20 0. 32 0. 18 0. 15 0. 07 0. 03 Comment: Any probability distribution for a discrete random variable X has the usual properties of a probability distribution:
1. 0 ≤ PX = x (^) i ≤ 1 for i = 1, 2, … , k
Definition The probability distribution for a discrete random variable is also called probability mass function or simply probability function.
1. What must be the probability of 7 people in a household for this to be a legitimate discrete distribution? 2. What is the probability that a randomly chosen household contains more than 5 people? 3. What is the probability that a randomly chosen household contains no more than 2 people? 4. What is P 2 ≤ X ≤ 4 ?
Definition The mean , or expected value of a discrete random variable X is denoted by EX and defined as
EX = (^) ∑ i= 1
k x (^) i Px (^) i ,
where, Px (^) i denotes PX = x (^) i .
Definition The variance of a discrete random variable X is denoted by σ^2 X^ and defined as
σ^2 X = (^) ∑ i= 1
k x (^) i − EX^2 Px (^) i ,
Definition The positive square root of the variance of X is called the standard deviation of X and is denoted by σ X = σ^2 X
Definition The probability distribution of a continuous random variable X is a function denoted by fx, such that PX takes on values in a set A = the area under the function fx above the set A. The function fx must satisfy
1. fx ≥ 0, ∀x (for all x ) 2. The total area under fx = 1_._ Example Let X be the length of a pregnancy in days, with X~N266, 256. What is the probability that a pregnancy lasts at least 310 days?
