Conditional probability is a statistical measure that calculates the likelihood of an event occurring based on the occurrence of a previous event. Specifically, it is the probability of a particular outcome given that another event has already occurred.
It can be expressed mathematically as:
P(A|B) = P(A ∧ B)/P(B)
Where P(A) represents the probability of event A occurring, P(B) is the probability of event B occurring, P(A|B) is the probabiity of event A occuring given evnet B occured, and P(A ∧ B) is the probability of both A and B occurring.
For example, consider a bag that contains three red marbles and two blue marbles. If we randomly draw one marble from the bag, the probability of selecting a red marble is 3/5 or 0.6. However, if we draw a marble from the bag and know that it is blue, the probability of the next marble being red becomes conditional on the first result.
If we drew a blue marble in our first draw and then replaced it, the probability of drawing a red marble on the second draw would be:
P(Red|Blue) = P(Red ∧ Blue)/P(Blue)
Since the marble was replaced after the first draw, P(Blue) = 2/5. The probability of drawing a red and blue marble is 3/5 * 2/5 = 6/25. Therefore:
P(Red|Blue) = (3/5 * 2/5)/(2/5) = 3/5
This means that the probability of drawing a red marble on the second draw, given that we drew a blue marble on the first draw, is 0.6.