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OVERVIEW OF THE TOPIC

Definition:  In simple language, it is the probability of an event occurring given that the second event has already happened.

Let's understand that in detail:

Suppose there are two events,

• Event A
• Event B

Now we have to find out that event A will occur or not given that event B has already happened.

The meaning of conditional probability is one of the most basic and one of the most important in probability theory.  But conditional probability can be quite tricky and require careful interpretation.  For example, there need not be a proper relationship between A and B, and they don't have to occur simultaneously.

Conditional probability is the probability of a second event occurring given that a first event has already happened. Conditional probability with dependent events are when one event affects the outcome of another event in a probability scenario.

Now let's take a quantitative example:

Suppose there are 5 Red balls and 7 blue balls in a bag. Now if we have to find the probability of picking up 1 ball at random which will be blue. Then it will be an example of simple probability. But what if we have to find the probability of picking up 2 balls simultaneously being both blue. Now here comes the concept of Independent events and dependent events.

In first probability picking a single blue ball is not dependent on any of the event, so that's why this type of event is known as an independent event. But in the second probability, picking of the second blue ball is dependent on the event 1 which is picking the first blue ball. This type of event is known as dependent events.

So Conditional probability is known as the probability of dependent events.

Once Albert Einstein quoted that "Nothing is absolute, everything is connected, and everything is relative." And so, in every aspect of life, we will need the probability of such type of events to smoothen the further solution. This type of probability is known as a conditional probability.

Thus, the conditional probability is the probability of dependent events as in relevant events. And it is greatly used everywhere. Let me give you one more real-life example;

Let's say we have to find the probability of finding a postgraduate man out of a city. So this is the example of conditional probability. Because our final probability of an event occurring of being post-graduate is dependent on being a man.

Now let's take an example of a card game, suppose a player wants to draw two cards of the same suit to win. Of the total 52 cards, there are 13 cards for each suit. Suppose the playerfirst draws a club. The player now wishes to draw a second club. Since one club has already been chosen, there are currently 12 clubs remaining in a deck of 51 cards. So, the conditional probability P (Draw second club/ First card a club) = 12/51.

Suppose an individual is trying to get admission to a college determines that he has a 70% chance of being accepted, and he knows that dormitory housing will only be provided for 50% of all of the accepted students. The possibility of the student being accepted and getting dormitory housing is defined by P (Accepted & Dormitory Housing) = P (Dormitory Housing/Accepted) P (Accepted) = (0.50)*(0.70) = 0.35.

The conditional probability of a given event B concerning a given event A is the probability that event B occurs while given that event A has already happened.

The Venn diagram given below describes P(A), P(B), and P (A and B). 2 parts have to be divided to find P(B|A). The concept of conditional probability is closely connected to the ideas of independent and dependent events. Probability problems that give knowledge about the outcome can often lead to unique results. An excellent example of this is one's initial intuition is incorrect on these kinds of problems. However, in-depth knowledge of conditional probability can help one obtain the correct result.

Suppose two players, often called Alex and Robin, initially have x and T - x dollars, respectively. A ball, which is red with probability p and black with probability q = 1 - p, is drawn from a bag. If a red ball is drawn, Alex must pay Robin one dollar, while Robin must pay Alex one dollar if the ball drawn is black. The ball is replaced every time, and the game continues until one of the players is destroyed. It is quite complicated to determine the probability of Peter's ruin by direct analysis of all possible cases. In this type of examples, the conditional probability is greatly used.

Summary: So conditional probability is one of the essential concepts of probability, and it is used everywhere in the real-time scenarios. The original idea of conditional probability is independent events and dependent events.

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• Independence of Random Variables
• Fluctuations in Coin Tossing and Random Walks
• Conditional Probabilities
• Joint and Conditional Distributions
• Occupancy and Matching Problems
• Independence
• Bayes' Rule
• Applied Probability
• Measure Theoretic Probability
• Probability Theory
• Random Variables
• Combinatorial Probability
• Standard Discrete Distributions: Uniform, Binomial, Poisson, Geometric, Hypergeometric
• Statistics
• Markov chains
• Estimation Theory
• Expectation and Integration
• Combination of Events
• Continuous Random Variables
• Equiprobability
• Heuristics in Judgment and Decision-Making
• Tchebychev's Inequality
• Standard Univariate Densities
• Estimators
• Conditional expectation
• Discrete distributions: Expectations, Variance and moments
• Probability Spaces and Probability Measures
• Transformation of Variables
• Convergence of Random Variables
• Martingales
• Inverse probability
• Probability Generating Functions
• Moment Generating Functions
• Laws of Large Numbers

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