Difference between Joint probability and Conditional probability

Joint probability and conditional probability are two concepts in probability theory that deal with the likelihood of events, but they are used in different contexts and measure different things.

Joint Probability

• Definition: Joint probability is the probability of two events happening at the same time.
• Notation: $( P(A \cap B) ) or ( P(A \text{ and } B) )$
• Example: The probability of drawing a King and then a Queen from a deck of cards in two draws (without replacement) is an example of joint probability.

Conditional Probability

• Definition: Conditional probability is the probability of an event occurring given that another event has already occurred.
• Notation: $( P(A|B) )$
• Example: The probability of drawing a Queen given that a King has already been drawn (without replacement) from a deck of cards.

Key Differences

Context:
– Joint probability considers the likelihood of two events happening together.
– Conditional probability considers the likelihood of one event happening given that another event has already happened.

Formula:
– Joint Probability: $( P(A \cap B) )$
– Conditional Probability: $( P(A|B) = \frac{P(A \cap B)}{P(B)} )$

Interpretation:
– Joint probability measures the combined likelihood of both events.
– Conditional probability measures the likelihood of one event in the context of the occurrence of another event.

Example to Illustrate Both Concepts

1. Joint Probability Example:
   • Drawing a King and then a Queen from a deck of cards without replacement.
     $( P(\text{King and Queen}) = \frac{1}{13} \times \frac{4}{51} = \frac{4}{663} ).$
2. Conditional Probability Example:
   • Drawing a Queen given that a King has already been drawn.
     $( P(\text{Queen}|\text{King}) = \frac{4}{51} )$

In summary, joint probability looks at the combination of events happening together, while conditional probability focuses on the probability of an event occurring in the context of another event having already occurred.