Theoretical probability is a way to predict how likely an event is to occur using mathematics rather than experiments. It assumes that all outcomes are equally probable. To find it, we divide the number of favorable outcomes by the total number of possible outcomes:
 

\(P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

It helps us understand the theoretical outcomes when they are known and predictable.

For example, if you roll a fair six-sided die, what is the probability of getting a 4?

Here, 

Favorable outcome = 1

Possible outcomes = 6

So, the probability of getting 4 = \(1 \over 6\)
 

Therefore, the theoretical probability of rolling a 4 is \(1\over6\)

Probability is the way to measure how likely an event is to happen. The probability is always between 0 and 1. If the probability is 1, the event is sure to occur; if it is 0, the event cannot happen. There are two main types of probability: experimental and theoretical.

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.
 

Theoretical Probability helps us predict how likely an event is to occur without doing any experiments. Theoretical Probability is the ratio of the number of favorable outcomes to the total number of possible outcomes. It is written as:

\(\text{Theoretical Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}} \)

Theoretical probability tells us how likely an event is to happen by using theory instead of doing an experiment. To calculate it, follow these simple steps:

• Understand the situation or experiment
• List and count all the possible outcomes
• Identify and count the favorable outcomes
• Use the formula: 

\(\text{Theoretical Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}} \)

For example, a person owns 30 of 500 total raffle tickets. Find the probability of winning. 
 

Here, the favorable outcomes = 30

Total possible outcomes = 500

p(winning) \(= {{30\over 500 }}= 0.06\)

The theoretical probability of winning the raffle is 0.06 or 6%.

Theoretical probability is used to understand the chances of an event occurring by reasoning rather than experimentation. Here are a few tips and tricks to master theoretical probability. 

• Memorize the formulas: Theoretical probability: \(P(E) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} \). 
   

• Remember that probability is always the number of favorable outcomes divided by the total number of possible outcomes.
   

• Teachers can use real-life objects such as coins, dice, and cards to help students better understand theoretical probability.
   

• Parents can help children understand theoretical probability by using everyday situations, such as predicting which fruit will be picked first from a bowl.
   

• Students can use diagrams, such as tables or tree diagrams, to find outcomes more clearly.

Theoretical probability is used in different fields to predict outcomes and make informed decisions. Here are some real-life applications of theoretical probability. 
 

• Gambling and Games of Chance: Theoretical probability is used to predict outcomes in games like roulette, dice, and card games. It can also be used to calculate the odds of winning. For example, the probability of rolling a four on a fair six-sided die is: p(4) = \(1 \over 6\).
   
• Insurance and Risk Assessment: Theoretical probability is used in insurance and risk assessment to forecast stock price movements. For example, an insurance company may calculate the probability that a 30-year-old will file a health claim in a year using past data, then use that probability to set premium amounts.
   
• Traffic Management and Safety: Traffic engineers use theoretical probability to assess accident risk, optimize signal timing, and alleviate congestion.
   
• Lottery and raffles: Probability helps in estimating the chances of winning based on the number of entries. 
   
• Quality Control in Manufacturing: Factories use probability to estimate the chance of producing defective items and improve production processes. For example, if a machine produces one defective product out of every 500 items, then p(defective) = \(1\over 500\).