Probability Basics

Probability is the certainty of an event taking place.

Conditional Probability

Assuming that B has been observed, the following tells us about the probability that A will take place:

P(A given B) = P(A and B)/P(B)

That is, we calculate the ratio of number of times both A and B occurred, and, the number of times B has occurred.

For short, we write:

P(A|B) = P(AB)/P(B)

where P(A|B) is the conditional probability, P(AB) is the joint, and P(B) is the marginal.

If we have more events, we can use chain rule: P(ABC) = P(A|BC).P(AB|C).P(C)

If A is not dependent on B, then P(AB) = P(A).P(B)

Let me walk you through some examples:

1. Assume we have a box with 3 red balls and 1 blue ball. What is the probability of drawing red balls in the first 2 trails?

By chain rule,

P(B1=r,B2=r) = P(B2=r|B1=r).P(B1=r) = (2/3).(3/4) = 1/2


P(B1=r,B1=r) is the probability of drawing a red ball in the first trial and picking the second ball in the next try.

P(B2=r|B1=r) is the probability of drawing the red ball in the second trail given having already picked the red ball in the first try.

P(B) is the probability of drawing red ball.

2. What is the probability that the 2nd ball drawn from a set {r,r,r,b} will be red?

Using marginalization, P(B2=r) = P(B1=r, B2=r) + P(B1=b,B2=r) = 1/2 + 1/4 = 3/4

Alright that’s it for now! Thank you for spending your time. Cheers!

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