1. Meaning of Conditional Probability
Conditional probability measures the chance of an event happening after we already know that another event has happened. The idea is that the information about the first event changes the sample space for the second.
In simple words: it answers the question, “What is the probability of A happening if B has already happened?”
Examples:
- Finding the chance of drawing a king given that the drawn card is already known to be a face card.
- Finding the chance of picking a red ball when it’s known that the ball picked is from a particular box.
2. Definition and Formula
The conditional probability of event A given event B is written as \( P(A|B) \).
\( P(A|B) = \dfrac{P(A \cap B)}{P(B)} \)
This formula works when the probability of B is not zero. It reflects how the overlap between A and B compares to the total range allowed by B.
3. Understanding the Restricted Sample Space
When we are told that event B has already occurred, we only consider outcomes that belong to B. This creates a restricted sample space.
Conditional probability then measures how much of this restricted space also belongs to A.
3.1. Example
Suppose a card drawn from a deck is known to be red. What is the probability that it is a king?
Restricted space: all red cards = 26 cards.
Favourable outcomes: red kings = 2 cards.
\( P(\text{King | Red}) = 2/26 = 1/13 \)
4. Examples Using Formula
Let’s consider two events from a deck:
- A: drawing a queen
- B: drawing a face card
There are 4 queens and 12 face cards. Overlap (face card that is also a queen) = 4.
- \( P(A \cap B) = 4/52 \)
- \( P(B) = 12/52 \)
\( P(A|B) = \dfrac{4/52}{12/52} = 1/3 \)
5. When Conditional Probability is Used
Conditional probability is useful in many situations where events influence each other:
- Drawing cards without replacement
- Choosing balls from different boxes
- Detecting faulty items in production
- Finding chances using partial information
It helps describe how one event affects the likelihood of another.