Conditional Probability: Understanding Probabilities of Dependent Events

Explore conditional probability and how to calculate the probability of an event given that another event has already occurred. This guide defines conditional probability, explains the formula for calculating conditional probabilities, and provides examples to illustrate its application.

Conditional Probability in Discrete Mathematics

Definition of Conditional Probability

Conditional probability deals with the probability of an event happening given that another event has already happened. If A and B are two dependent events, the probability of A happening given that B has already happened is written as P(A|B) and calculated as:

P(A|B) = P(A ∩ B) / P(B)

Where P(A ∩ B) is the probability of both A and B occurring, and P(B) is the probability of B occurring.

Conditional Probability: Interchanging Events

Similarly, the probability of B occurring given that A has already occurred is:

P(B|A) = P(A ∩ B) / P(A)

Proof of the Conditional Probability Formula

Let S be the sample space. We know that P(A ∩ B) = P(A|B) * P(B). Rearranging this equation gives us the formula for conditional probability: P(A|B) = P(A ∩ B) / P(B).

Example: Drawing Cards

Let's consider drawing two cards from a standard deck of 52 cards without replacing the first card. What's the probability of drawing two hearts?

  1. First Draw (Event A): The probability of drawing a heart is P(A) = 13/52 = 1/4 (there are 13 hearts in a deck of 52 cards).
  2. Second Draw (Event B), given a heart was drawn first: After drawing one heart, there are 12 hearts left and 51 total cards. The probability of drawing a second heart given the first was a heart is P(B|A) = 12/51.
  3. Probability of both events: To find the probability of both events happening, we multiply their probabilities: P(A ∩ B) = P(A) * P(B|A) = (1/4) * (12/51) = 1/17.

Therefore, the probability of drawing two hearts without replacement is 1/17.

Conclusion

Conditional probability is a powerful tool for calculating probabilities when events are dependent. Understanding how to calculate and interpret conditional probabilities is essential in probability and statistics.