Conditional Probability

• Conditional Probability is defined as the probability of an event $A$ occurring, under the condition that event $B$ has already occurred. This is mathematically denoted as $P(A|B)$, which is read as 'the probability of $A$ given $B$'.

• The Restricted Sample Space is the most critical conceptual shift in conditional probability. Instead of considering all possible outcomes in the universal set $\xi$, we focus exclusively on the outcomes contained within the 'given' event $B$.

• The Formal Formula for calculating this value is $P(A|B) = \frac{P(A \cap B)}{P(B)}$, provided that $P(B) > 0$. This formula expresses the ratio of the intersection (where both $A$ and $B$ occur) to the total probability of the condition $B$ itself.

• The Multiplication Rule is derived directly from the conditional probability formula, stating that $P(A \cap B) = P(B) \times P(A|B)$. This principle allows us to find the probability of two events happening in sequence by multiplying the first event's probability by the second's conditional probability.

• Successive Events often involve conditional probability when the outcome of the first event changes the conditions for the second. This is most commonly seen in 'sampling without replacement' scenarios where the total number of items decreases after each draw.

• Independence occurs when the knowledge that $B$ has happened does not change the probability of $A$. Mathematically, events $A$ and $B$ are independent if $P(A|B) = P(A)$, meaning the condition provides no new information.

• It is vital to distinguish between Joint Probability $P(A \cap B)$ and Conditional Probability $P(A|B)$. Joint probability considers the chance of both occurring relative to the entire sample space, whereas conditional probability considers the chance of $A$ relative only to the subset $B$.

| Feature | Joint Probability $P(A \cap B)$ | Conditional Probability $P(A|B)$ | | --- | --- | --- | | Denominator | Total outcomes in $\xi$ | Outcomes in event $B$ | | Perspective | 'Both happen' | 'A happens, given B' | | Formula | $P(A) \times P(B|A)$ | $\frac{P(A \cap B)}{P(B)}$ |

• Another critical distinction is between Independent and Dependent events. In independent events, $P(A|B) = P(A)$, while in dependent events, the occurrence of $B$ alters the likelihood of $A$, meaning $P(A|B) \neq P(A)$.

• Identify Keywords: Always look for phrases like 'given that', 'if it is known that', or 'of those who...'. These phrases signal that the denominator of your probability fraction must be restricted to a specific subgroup.

• Tree Diagram Denominators: When drawing tree diagrams for sampling without replacement, remember to reduce both the numerator and the denominator for the second set of branches if the same category is selected twice.

• Sanity Check: A conditional probability $P(A|B)$ can never be greater than 1, and it is usually different from $P(B|A)$. If your calculation for $P(A|B)$ results in a value larger than $P(A \cap B)$, this is expected because you are dividing by a number less than 1 ($P(B)$).

• Confusing $P(A|B)$ with $P(B|A)$: This is known as the 'prosecutor's fallacy'. The probability of being a doctor given you are in a hospital is very different from the probability of being in a hospital given you are a doctor.

• Forgetting to Update Totals: In multi-stage problems without replacement, students often forget to subtract 1 from the total population for the second event. This leads to using the original sample space size instead of the restricted one.

• Misinterpreting Independence: Students often assume events are independent unless told otherwise. Always check if the first event physically or logically changes the conditions for the second before assuming $P(A|B) = P(A)$.