Question

What is difference between joint probability and conditional probability

Answer 1

A conditional probability is the probability of an event X occurring when a secondary event Y is true. Mathematically, it is represented as P(X | Y). This is read as “probability of X given/conditioned on Y”.

For example, if someone asked you the probability of getting a diamond with the G color, P(X=G), we can use Table 3 to find the marginal probability of this event. But what if you had an additional layer of information where you knew that the diamond was also of ideal cut? This becomes a conditional probability since we have an event that is already true. A conditional probability can be calculated as follows:

P(X | Y)=P(X,Y)P(Y)

Recall that the marginal probability is simply summing up the joint probabilities while holding one variable constant. So we can further breakdown this equation as follows:

P(X | Y)=P(X,Y)∑x∈SXP(X=x,Y)

So for us to work this out for our particular question, we need two pieces of information:

P(Y=ideal): Marginal probability of Y = ideal.

P(X=G,Y=ideal): Joint probability of X = G and Y = ideal.

So we can calculate the conditional probability as follows:

P(X=G | Y=ideal)=P(X=G,Y=ideal)∑x∈SXP(X=x,Y=ideal)

joint

mathematical equation for joint probabilities which actually uses both the conditional and marginal probability equations. Starting with the conditional probability equation, we can do a bit of algebraic manipulation for defining joint probabilities now:

P(X | Y)P(X | Y) P(Y)P(X | Y) ∑x∈SXP(X=x,Y)=P(X,Y)P(Y)=P(X,Y)=P(X,Y)