How should we understand, interpret and use probability? This post will explain what probability is and how probabilities are calculated...
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Consider the following example: For the academic year 2020-21, the total number of students enrolled in undergraduate programs in the United States was 15.9 million, with females accounting for 58% of the total.
What is Probability?
In simple words, the probability of an event is defined as the proportion of times an event occurs over many repetitions.
In our example, the event is that the enrolled undergraduate student is a female. There were 15.9 million repetitions of the conditions. And the event itself occurred 58% of the time, meaning there were roughly 9.2 million female undergraduate students enrolled for the year 2020-21.
We must take into account that the probability given here is a result of the experiment being repeated many, many times. This means that it’s not always applicable to individual events.
Actual or objective probabilities should be distinguished from subjective probabilities, which are not based on experiments and may result in different probabilities for the same event. Someone thinking that "the probability that my best friend will call today is 30%" is an example of a subjective probability because it is based on his/her personal feelings.
But a probability is not the same as a prediction. For example, it would be a mistake if someone learns about the undergraduate enrolled student's gender probability and then predicts that they will be admitted because they are female.
Probabilities alone are neither subjective guesses nor predictions of future events, they are simply descriptive.
Note: This is not to say that probabilities aren’t used to help inform prediction. They do, although we’ll need some more sophisticated statistical methods before we can do that.
Rules of Probability
The probability of an event lies between 0 and 1, inclusive; it cannot be less than 0 or greater than 1. If probability is 0, the event never occurs. If the probability is closer to 0, the more improbable it is that the event will occur. If the probability is 1, the event will always happen. If the probability is closer to 1, the more certain it is that the event will happen. A few simple rules, which we will discuss below, are used to calculate the probability of an event.
Complement Rule:
The Complement Rule states that the sum of the probability of an event A happening (P(A)) and its complement or the event not happening (P(~A)) is always equal to 1.
P(A) + P(~A) = 1
P(~A)= 1- P(A)
In our example, we saw that the probability that an undergraduate student is a female is 58%. So, what is the probability that the undergraduate student is a male?
P(Female undergraduate student) = 9.2 million/ 15.9million = 0.58
P(Male undergraduate students)= 1- P(Female undergraduate student)
= 1- 0.58
= 0.42
Equally Likely Outcomes Rule:
The Equally Likely Outcomes Rule states that if there are n equally likely outcomes of an event A, of which one is called a success, then the probability of success is
P(A) = number of ways Event A can occur (n) / total number of possible outcomes (s)
For example, the probability of guessing the birthday of one’s best friend correctly, not taking leap-years into account is
P(Guessing Best Friend's Birthday)= 1 / 365
In this case, the number of ways one can guess the correct birthday can be only 1, and the total number of ways from which we have to select is 365 (as there are 365 days in a non-leap year).
Addition Rule:
Two events are known to be mutually exclusive from each other when they cannot occur at the same time.
In this case, the addition rule states that in order to calculate the probability of Event A happening or Event B happening, we can add them.
P(A or B)=P( A) + P B)
For example, what is the probability of getting a head P(H) or tail P(T) when a coin is flipped?
P(H)= 1/2
P(T)= 1/2
P(H or T) =P(H) + P(T)
=1/2 + 1/2 = 1
Multiplication Rule:
Two events are said to be independent of each other when the occurrence of one event does not impact the occurrence of other events at all. In this case, the multiplication rule states when the two events are independent of each other and mutually exclusive then the probability of occurrence of Event A happening and Event B happening is the multiplication of probability for both events.
P(A and B)=P(A) x P(B)
For example, if a dice is thrown two times, what is the probability that the first time we will get an odd number and the second time we will get an even number?
P(Odd)=3/6
P(Even)=3/6
P(Odd and Even)=3/6 x 3/6 = 1/4
Real Life Probability Applications
Probability is used in all aspects of real life, including finance, medicine, sports, and investing.
Example 1: Credit Card Company
Credit card companies frequently use probability to determine how likely it is that certain individuals will pay their credit card bill on time. They decide whether or not to issue a credit card based on this information.
Example 2: Sports
Sportsbooks rely heavily on probability to establish betting lines for upcoming games. Taking into account both teams' recent performances, a sportsbook might assign a 20% chance of victory to team A and an 80% chance of victory to team B. Given these odds, the company will likely increase the payout for customers who wager on team A to win.
Example 3: Staffing needs
Probability is frequently used by department stores when deciding how many employees to schedule for a given day. A department store might use a model that predicts that there is an 80% chance they will see more than one thousand customers on Black Friday if they advertise heavily. In order to accommodate the expected influx of customers, a specific number of staff members will be scheduled to work that day.
Key Takeaways:
The probability of an event is defined as the proportion of times it occurs over many repetitions.
Probability should not be confused with prediction, because prediction requires additional statistical information.
The probability range is always between 0 and 1.
The complement rule, equally likely outcomes rule, addition rule, and multiplication rule are used to calculate probability.
Practical applications of probability can be found in a wide variety of fields, from economics and medicine to sports and investing.
References:
Walter, G. (2022). Introduction to Statistics [MOOC]. Coursera. https://www.coursera.org/learn/stanford-statistics
Walter, G. (2022) Probability [MOOC lecture]. In Walter, G., Introduction to Statistics. Coursera (Stanford University). https://www.coursera.org/learn/stanford-statistics/lecture/9LeSM/theinterpretation-of-probability
Zach. (2021, November 4). 10 Examples of Using Probability in Real Life. Statology. https://www.statology.org/probability-real-life-examples/