Day - 3 Probability in Machine Learning

Conditional Probability

The?conditional probability, as its name suggests, is the probability of happening an event that is based upon a condition.

For example, assume that the probability of a boy playing cricket in the morning is 95% (0.95) whereas the probability that he plays given that it is a rainy day is less which is 10% (0.1). Then the former case is just normal probability whereas the latter case is the conditional probability.

What is Conditional Probability?

The "probability of A given B" or the "probability of A with respect to the condition B" is denoted by the conditional probability P(A | B) or P (A / B). Thus, P(A | B) represents the probability of A which happens after event B has happened already. the probability of an event may alter if there is a condition given.

Definition of?Conditional Probability

If A and B are two events associated with the same sample space of a random experiment,?the conditional probability of event A given that B has occurred is given by P(A/B) = P( A ∩ B)/ P (B), provided P(B) ≠ 0.

Formula of Conditional Probability?as follows :

P(A | B) = P(A ∩ B) / P(B)?(Note that P(B) ≠ 0 here)

Similarly, we can define P(B | A) as follows :

P(B | A) = P(A ∩ B) / P(A)?(Note that P(A) ≠ 0 here)

These formulas are also known as the "Kolmogorov Definition" of conditional probability.

Here:

• P(A | B) = The probability of A given B (or) the probability of A which happens after B
• P(B | A) = The probability of B given A (or) the probability of B which happens after A
• P(A ∩ B) = The probability of happening of both A and B
• P(A) = The probability of A
• P(B) = The probability of B

Derivation of Conditional Probability

P(A/B) = Number of events favorable to A ∩ B ÷ Number of events favorable to B So it can also be written as :

Thus P(A | B) = P(A ∩ B) / P(B)

Bayes Theorem

Bayes theorem is a theorem in probability and statistics, named after the Reverend Thomas Bayes, that helps in determining the probability of an event that is based on some event that has already occurred.

What is Bayes Theorem?

Bayes theorem, in simple words, determines the conditional probability of event A given that event B has already occurred based on the following :

• Probability of B given A
• Probability of A
• Probability of B

Here :

• P(A) = how likely A happens(Prior knowledge)- The probability of a hypothesis is true before any evidence is present.
• P(B) = how likely B happens(Marginalization)- The probability of observing the evidence.
• P(A|B) = how likely A happens given that B has happened(Posterior)-The probability of a hypothesis is true given the evidence.
• P(B|A) = how likely B happens given that A has happened(Likelihood)- The probability of seeing the evidence if the hypothesis is true.

Central Limit Theorem

Central Limit Theorem says that the probability distribution of arithmetic means of different samples taken from the same population will closely resemble a normal distribution. In general, for the central limit theorem to hold, the sample size should be equal to or greater than 30.

Central Limit Theorem Definition

The central limit theorem states that irrespective of a random variable's distribution if large enough samples are drawn from the population then the sampling distribution of the?mean?for that random variable will approximate a normal distribution. This fact holds true for samples that are greater than or equal to 30. In other words, as more large samples are taken, the graph of the sample means starts looking like a normal distribution.

Central Limit Theorem Formula

Thus, the central limit formula says that the random variable of the sample means will be normally distributed with a mean that will be equal to the original distribution and?standard deviation?given by σ / √n.