Probability

What is the meaning of Probability? 

The meaning of Probability is quite simple. It is defined as the ratio of favorable outcomes by the total number of possible outcomes, where the event has an ‘n’ number of outcomes. If the number of favorable outcomes is denoted by ‘x,’ then the formula of Probability of an event is given as: 

Probability (event) = favorable outcomes / total outcomes = x / n 

Terminologies in Probability 

• Experiment: An experiment in the Probability of an event is defined as the trial conducted to produce an outcome
• Sample: All the predicted possible outcomes before the start of the trial are called a sample. For example, the head and tail are the sample space for a coin toss experiment
• Favorable Outcome: The desired result in an experiment is called the favorable outcome
• Random Experiment: Any experiment that has a fixed set of outcomes, like tossing a coin to get head or tail, is called a random experiment
• Equally Likely Events: The meaning of Probability in Equally Likely Events is that the events have equal chances of occurring again and again
• Exhaustive event: It occurs when the set of all outcomes of an experiment becomes equal to the sample space

Formulas of Probability

The probability of an event is formulated by the ratio between favorable and total outcomes. It can be expressed as, 

P(A) = number of favorable outcomes to A / Total number of possible outcomes

Where P(B) = Probability of an event B

n(B) = favorable outcomes of B 

n(S) = total number of events in sample space S 

There are also other formulas related to the probability of an event. 

• Additional Probability: It is the union of two events, A and B, which is expressed as: 

P(A or B) = P(A) + P(B) – P(A∩B)

P(A ∪ B) = P(A) + P(B) – P(A∩B)

• Conditional Probability: Conditional Probability occurs when A has already occurred, and the probability of B is desired, or vice versa. It is also given as P(B, given A) = P(A and B), P(A, given B)

P(B∣A) = P(A∩B)/P(A)

• Complementary Probability: It occurs when two events are complementary to each other. P(not A) = 1 – P(A) or P(A’) = 1 – P(A)

P(A) + P(A′) = 1.

• Multiplication Probability: This probability happens when an event is the intersection of two other events, which means that two events need to occur simultaneously

           P(A and B) = P(A)⋅P(B).

          P(A∩B) = P(A)⋅P(B∣A)

Types of Probability

• Classical Probability: It is also known as theoretical probability. It basically states that, if there are B equally likely outcomes, and another event X has A outcomes, then the probability of an event X is A/B. For example, when you roll a fair die, there are 6 equally likely outcomes and you get a probability of ⅙
• Empirical Probability: Here, the probability of an event is evaluated through thought experiments
• Subjective Probability: This probability is based on an individual’s belief in an event taking place. For example, the opinion of a cricket fan is dependent on the faith of his team winning the match rather than a mathematical calculation
• Axiomatic Probability: The axioms by Kolmogorov are applied to all the types of probability. The applications provided by these axioms quantify the chances of occurrence of an event. The applications state that the smallest probability is zero and the highest is one. It also states that a certain event has a probability equal to one and any two mutually exclusive events cannot occur at the same time

Bayes’ Theorem on Conditional Probability 

The theorem of Conditional Probability is based on the occurrence of other events and it helps in calculating the probability of an event based on the occurrence of the other events. 

For example, if there are 3 bags with each bag having blue, green, and yellow balls, the probability of picking a yellow ball from the third bag depends on the other balls in the bag. This is Conditional Probability, which is given as: 

P(A|B) = P(B|A)⋅ P(A) P(B)

Where P(A|B) is the frequency of event A happening on the condition that B is happening. 

P(B|A) is the frequency of event B happening on the condition that A is happening. 

P(A) and P(B) are the probabilities of A and B, respectively. 

Conclusion

Probability is the measure of an event going to happen, and it is always represented as a fraction. It always lies between 0 and 1, and a random experiment cannot predict the exact outcomes.