Conditional Probability
Conditional probability is a cncept in probability theory that describes the likelihood of an event occurring given that another event has already occurred. It's denoted as P(A|B), which reads as "the probability of A given B".
The formal definition of conditional probability is:
$P(A|B) = \frac{P(A \cap B)}{P(B)}$
Where:
- P(A|B) is the conditional probability of A given B
- P(A ∩ B) is the probability of both A and B occurring
- P(B) is the probability of B occurring
Let's consider a deck of 52 cards. We want to find the probability of drawing a king given that we've drawn a face card.
- P(King) = 4/52 = 1/13
- P(Face card) = 12/52 = 3/13
- P(King and Face card) = 4/52 = 1/13
P(King | Face card) = P(King and Face card) / P(Face card) = (1/13) / (3/13) = 1/3
So, the probability of drawing a king, given that we've drawn a face card, is 1/3.
The conditional probability formula can also be rearranged to: P(A ∩ B) = P(B) * P(A|B) This form is particularly useful when we know the conditional probability and want to find the probability of both events occurring.
Independent Events
Two events A and B are considered independent if the occurrence of one event does not affect the probability of the other event occurring. Mathematically, this is expressed as:
$P(A|B) = P(A)$
This means that the probability of A occurring remains the same whether or not B has occurred.
For independent events, we can also say:
$P(A \cap B) = P(A) * P(B)$
Consider rolling a fair six-sided die twice. The outcome of the first roll doesn't affect the probability of any outcome on the second roll.
Let A be "rolling a 3 on the first roll" and B be "rolling an even number on the second roll".
P(A) = 1/6 P(B) = 3/6 = 1/2
P(A ∩ B) = P(A) * P(B) = 1/6 * 1/2 = 1/12
We can verify that A and B are independent: P(B|A) = P(B) = 1/2
Testing for Independence
To test whether two events are independent, we can use the following methods:
- Check if P(A|B) = P(A)
- Check if P(B|A) = P(B)
- Check if P(A ∩ B) = P(A) * P(B)
If any of these conditions are true, the events are independent.
Let's test for independence in a scenario where we have a bag with 5 red marbles and 5 blue marbles. We draw two marbles without replacement.
Let A be "first marble is red" and B be "second marble is blue".
P(A) = 5/10 = 1/2 P(B) = 5/9 (because after drawing the first marble, there are 9 left)
P(A ∩ B) = (5/10) * (5/9) = 25/90
P(B|A) = 5/9
Since P(B|A) ≠ P(B), these events are not independent.
Students often confuse independent events with mutually exclusive events. Independent events can occur together, while mutually exclusive events cannot. For example, rolling a 6 and rolling an even number are not mutually exclusive, but they are independent.
