Probability Line

The probability line is a line that showcases the probabilities and how these probabilities are associated with each other. Seeing that the probability of an event is a number from 0 to 1, we can also use the probability line for the purpose of displaying the possible ranges of probability value. The line depicts that if an event is sure to happen, it will have a probability of 1. For example, the probability that it will rain at least once a year in Delhi is 1.

Number Line in Probability

The number line in probability displays that if an event will never occur or cannot occur, it will have a probability of 0.

For example, the probability that you can pick a yellow ball from a bag containing 7 green balls and 3 red balls is 0.

Game Theory in Probability

This theory is a part of probability distribution. This theory plays a major role in the decision-making process. It determines how a player moves and what his/her mind thinks. 

Zero Sum Game Theory in Probability

There are closed games in which the outcome is fixed. The resources of this game can neither be decreased or increased. The total benefit is always zero. The result of this game is that one wins and the other always loses.

Whereas when one player’s game does not correspond to another player’s loss then it is termed as a non zero sum game.

Examples of Probability Game Theory

Prisoner’s Dilemma

One of the best examples to understand game theory in a practical way is the prisoner’s dilemma. There are two prisoners in two separate cells. Both of them can be imprisoned for a minor offence, but not for a major offence unless they testify against each other. Following are the possibilities –

1.   If both of them confess, they will be imprisoned for a major offence and get 9 years in prison.

2.   If none of them confesses, they will be imprisoned for a minor offence and get only 1 year in prison.

3.   If prisoner 1 confesses and prisoner 2 does not then prisoner 1 will get 9 years in prison whereas prisoners 2 will get only 1 year.

4.   If prisoner 2 confesses and prisoner 1 does not then prisoner 2 will get 9 years in prison whereas prisoner 1 will get only 1 year.

 Game theory suggests that both of them should not confess and that’s how they will get minimum imprisonment.

Solved Examples

Example1:

Four students – A, B, C, and D are sitting in random order next to each other. Find the probability that B sits at the northeast corner of the room.

Solution:

To make 4 students sit at 4 corners of the room, there are 24 different ways. B sits at the northeast corner of the room – there are 6 different ways to it.

Thus, the required probability is = 6/24 = ¼

Example2:

Identify the probability of getting at least 1 heads, when the coin is thrown three times.

Solution:

Sample Space: HHH, TTT, TTH, HTT, THT, THH, THH, HTH = 8

Required Probability = 1-1/8 = 7/8

Example3:

12 People are sitting together at a table. Find out how 2 particular people are sitting next to each other?

Solution:

12 people can sit in 11 different ways.

No. of ways in which 2 people can sit together are = 10! * 2!

Required probability = 10! * 2! / 11! = 2/11

Quiz Time

Q1. What is a Zero-Sum Game?

The sum of losses to one player is equal to the sum of gains to others.

Q2. How Game Theory Models are Classified?

The division is done as per the number of players, the sum of all payoffs and the number of strategic. 

Q3. How is a Game Fair?

A game is fair if both upper and lower values of the game are the same and zero. 

Q4. What Happens When the Maximum and Minimum Values of the Game are the Same?

If the maximum and minimum values of the game are the same, then the saddle point will exist.

Q5. How Can a Mixed Strategy Game Be Solved?

It can be solved by an algebraic method, matrix method and graphical method.

Q6. When No Saddle Point is Found in a Payoff Matrix of a Game. How is the Value of the Game Found?

Reducing the size of the game to apply the algebraic method.