Theoretical Probability

The objective of applying theoretical probability is to predict the likely outcome of an event. The prediction is based on mathematical reasoning and known outcomes, and not on actual experiments. In this process, we assume that all outcomes are equally likely. This technique is used when the outcomes are well-defined and predictable.  
 
 

Make note of the following steps to find theoretical probability.


Step 1: Make sure the experiment that you are working on is well-defined. Determine the sample space by making a list of every experiment result.
 

Step 2: Determine which results satisfy the requirements of the event considered.


Step 3: Count all possible outcomes in the sample space.


Step 4: Count all favorable outcomes.
 

Step 5: Apply the formula:
P(E) = Number of Favorable Outcomes ÷ Total Number of Possible Outcomes.


Step 6: Put the probability in percentage or decimal or fraction format. 

Theoretical probability is calculated using logic and known outcomes, while experimental and empirical probabilities rely on real-world trials and observed data, with empirical focusing on long-term observations.
 

Here are some tips and tricks to master theoretical probability:
 

• Know your basics: Memorize the key formulas:

           Theoretical probability: P(E) = Favorable outcomes/Total Outcomes
           Complement Rule: P(Not A) = 1 − P(A)
 

• Build sample spaces efficiently: Use tree diagrams for sequences of events, grids or tables for dice and cards, and Venn diagrams for overlapping sets.
  
   
• Estimate before you calculate: Quickly estimate what the probability should be. If your answer is above 1 or negative, you know that there is a mistake.
  
   
• Understand the formula: Remember that theoretical probability = (Number of favorable outcomes) \(÷\) (Total number of possible outcomes).
  
   
• Use simple examples: Practice with coins, dice, or cards to quickly grasp the concept before moving to complex problems.

Theoretical probability can be used in many ways. Here are some examples:

 

• Gambling and Games of Chance: We use theoretical probability for predicting outcomes in games like roulette, dice, or cards. It can also be used to calculate the odds of winning.
  
  
   
• Insurance and Risk Assessment: We use theoretical probability in insurance and risk assessment to forecast stock price movements. We also use it to assess the likelihood of returns or losses on investments.

• Traffic Management and Safety: We use theoretical probability in traffic management to estimate the probability of collisions at intersections. We also use it to manage traffic signals and reduce congestion. It is also applied by self-driven vehicles to make decisions on the road. 
  
   
• Game outcomes: Calculating the chances of rolling a certain number on a die or drawing a specific card from a deck.
  
   
• Lottery and raffles: Estimating the likelihood of winning based on the total number of tickets sold.