Probability
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.
FAQs on Probability Line
1. What is a probability line in mathematics and how does it help in understanding the likelihood of events?
The probability line is a visual representation, similar to a number line, that shows the range of probability values from 0 (impossible event) to 1 (certain event). It helps students understand where an event lies between being impossible and certain, making it easier to evaluate and compare the likelihood of different outcomes in probability problems.
2. How can you use the probability line to compare the chances of two different events?
The probability line allows you to place the probabilities of two events on the same scale from 0 to 1. By comparing their positions, you can instantly see which event is more likely. For example, if Event A is at 0.25 and Event B is at 0.75 on the line, Event B is more likely to occur.
3. Can you give an example of representing a real-life scenario on the probability line?
If the probability of drawing a green ball from a bag of red and green balls is 0.2, and the probability of drawing a red ball is 0.8, both can be plotted on the probability line. This visual quickly shows the event with the higher likelihood, reinforcing the concept of relative probability.
4. What are some common misconceptions students have when using the probability line?
Common misconceptions include assuming that all events must be exactly at 0, 0.5 or 1, or misunderstanding that probabilities cannot be negative or greater than 1. Students may also confuse certainty (1) with high probability (close to but less than 1) and overlook rare events that still have a non-zero probability.
5. Why is game theory included in the probability chapter, and how does it relate to the probability line?
Game theory is included to model real-life decision-making scenarios where outcomes depend on probabilities and the strategies of different players. The probability line is used to visually analyze the range of possible chances for different strategies and aids in understanding concepts like zero-sum games and mixed strategies.
6. How do zero-sum and non-zero-sum games differ, and what is their importance in probability?
In a zero-sum game, one player's gain is exactly balanced by the losses of other players, so the total sum remains zero, while in a non-zero-sum game, the sum of outcomes can be positive or negative, allowing all players to gain or lose together. Understanding these helps interpret real-world scenarios in probability and strategic decision-making.
7. What steps are involved in solving a mixed-strategy game as per the CBSE 2025–26 syllabus?
To solve a mixed-strategy game, you typically:
- List possible strategies for each player.
- Set up a payoff matrix.
- Use algebraic, matrix, or graphical methods to find the best response strategies.
- Calculate equilibrium or optimal solutions where players randomize their choices.
8. How does the concept of saddle point relate to finding fair value in a game?
A saddle point occurs in a payoff matrix when the minimum of the row maxima equals the maximum of the column minima. If present, it means a fair value exists and both players know the best strategies, ensuring neither can improve their outcome unilaterally.
9. How is probability applied when solving typical board exam questions on arrangements or events?
Probability is applied by:
- Determining the sample space (all possible outcomes).
- Counting the number of favourable outcomes for the event in question.
- Dividing the number of favourable outcomes by the total number of possible outcomes.
10. In exam questions, how is the probability of compound events (like "at least one head in three coin tosses") calculated?
To find the probability of a compound event, use concepts like the complement or direct enumeration.
- Calculate the total number of possible outcomes.
- Find the number of outcomes that meet the condition (e.g., "at least one head").
- Probability = Number of favourable outcomes / Total outcomes.
11. Why is it crucial to understand the probability line when approaching CBSE Class 12 Maths board exams?
Mastering the probability line clarifies the range of possible outcomes for any given event, making it easier to analyze complex problems, avoid calculation errors, and correctly interpret board exam questions involving probability, game theory, or arrangements.
12. How can misinterpretation of sample space affect your answers in probability calculations?
A misinterpretation of the sample space—such as missing or duplicating outcomes—can lead to incorrect probabilities. This especially impacts complex arrangements or game theory questions in competitive and CBSE board exams, highlighting the importance of careful listing and counting of all possible outcomes.
