This lesson introduces the fundamental concepts of conditional probability, a key area in probability theory.
Definition of Conditional Probability
Conditional probability is the likelihood of an event occurring, given that another event has already occurred. The key idea is that the knowledge of a prior event can change the probability of a future event. For example, the probability of drawing a Queen from a deck of cards changes if you already know the card drawn is a face card.
Conditional Probability Notation: P(A|B)
The notation for conditional probability is $P(A|B)$. This is read as "the probability of event A occurring, given that event B has occurred."
- A is the event we are interested in.
- B is the condition, the event that we know has already happened.
The Conditional Probability Formula
The probability of A given B can be calculated using the following formula:
$P(A|B) = \frac{P(A \cap B)}{P(B)}$
Where:
- $P(A \cap B)$ is the probability that both event A and event B occur (the intersection of A and B).
- $P(B)$ is the probability that event B occurs. It is important that $P(B) > 0$, as we cannot condition on an impossible event.
Concept of a Reduced Sample Space
When we are given that event B has occurred, we are no longer considering all possible outcomes in the original sample space. Instead, our new, reduced sample space consists only of the outcomes where B is true. The conditional probability $P(A|B)$ is then the proportion of these outcomes in the reduced sample space that also satisfy event A.
Example: Suppose we roll a single six-sided die. The sample space is $S = \{1, 2, 3, 4, 5, 6\}$. Let A be the event 'the number is even' and B be the event 'the number is greater than 3'.
- If we want $P(A|B)$, we are given that the number is greater than 3. Our reduced sample space is now just $B = \{4, 5, 6\}$.
- Within this new sample space, which outcomes are even (event A)? The outcomes are $\{4, 6\}$.
- Therefore, $P(A|B) = \frac{\text{Number of outcomes in A and B}}{\text{Number of outcomes in B}} = \frac{2}{3}$.
Independent versus Dependent Events
The concept of conditional probability helps us define whether events are independent or dependent.
Independent Events: Two events A and B are independent if the occurrence of one does not affect the probability of the other. In this case:
$P(A|B) = P(A)$ and $P(B|A) = P(B)$.
Example: Flipping a coin twice. The outcome of the first flip does not change the probability of getting heads on the second flip.
Dependent Events: Two events are dependent if the occurrence of one does affect the probability of the other. In this case:
$P(A|B)
eq P(A)$.
Example: Drawing two cards from a deck without replacement. The probability of drawing a King on the second draw depends on whether a King was drawn on the first draw.
The General Multiplication Rule
By rearranging the conditional probability formula, we get the General Multiplication Rule, which is used to find the probability of the intersection of two events:
$P(A \cap B) = P(B) \cdot P(A|B)$
Or, equivalently:
$P(A \cap B) = P(A) \cdot P(B|A)$
This rule states that the probability of both A and B occurring is the probability of B occurring, multiplied by the probability of A occurring given that B has already occurred.
Example: What is the probability of drawing two Aces from a standard deck of 52 cards without replacement?
- Let A be drawing an Ace first, and B be drawing an Ace second.
- $P(A) = \frac{4}{52}$
- Given that the first card was an Ace, there are now 3 Aces left in a deck of 51 cards. So, $P(B|A) = \frac{3}{51}$.
- Using the multiplication rule: $P(A \cap B) = P(A) \cdot P(B|A) = \frac{4}{52} \cdot \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$.
