Conditional probability is a fundamental concept in probability and statistics that involves determining the probability of an event occurring given that another event has already taken place. This topic is crucial for understanding how probabilities change in light of new information and is a key area of study in the IGCSE Mathematics syllabus.

What is Conditional Probability?

Conditional probability, denoted as $P(A|B)$, represents the probability of event $A$ occurring after event $B$ has already occurred. The calculation of conditional probability is rooted in the relationship between these two events, focusing on their intersection and the overall probability of the conditional event.

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Fundamental Formula

The backbone of conditional probability is expressed by the formula:

where $P(A|B)$ is the probability of $A$ given $B$, $P(A \cap B)$ is the probability that both $A$ and $B$ occur, and $P(B)$ is the probability of $B$.

Utilising Venn Diagrams

Venn diagrams are instrumental in visualising and calculating conditional probabilities, showing the overlap (intersection) that represents the occurrence of both events.

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Example with a Venn Diagram

Consider a scenario with 60 students, where 30 study Maths, 25 study Physics, and 10 study both. We aim to find the probability of a student studying Physics given they study Maths.

• Step 1: Identify $P(M \cap P)$ or the joint probability of studying both Maths and Physics.
  • $P(M \cap P) = \dfrac{10}{60}$
• Step 2: Determine $P(M)$, the probability of studying Maths.
  • $P(M) = \dfrac{30}{60}$
• Step 3: Apply the conditional probability formula.
  • $P(P|M) = \dfrac{P(M \cap P)}{P(M)} = \frac{1}{3}$

Thus, the conditional probability is $\dfrac{1}{3}$.

Tree Diagrams in Conditional Probability

Tree diagrams excellently map sequential events, where each branch represents the outcomes and their respective probabilities.

Conditional Probability with a Tree Diagram

Imagine a bag containing 5 red and 3 blue marbles. Drawing two marbles successively without replacement, we determine the probability of the second marble being blue given the first is red.

Initial Probabilities

• $P(\text{First Red}) = \dfrac{5}{8}$
• $(P(\text{First Blue}) = \dfrac{3}{8})$

After the First Draw

• Drawing a blue marble after a red: $P(\text{Second Blue | First Red}) = \dfrac{3}{7}$

Thus, the conditional probability is $\dfrac{3}{7}$.

Practice Problems

Engage with these exercises to deepen your understanding of conditional probability.

Problem 1

A box contains 6 chocolate, 4 vanilla, and 2 strawberry ice creams. What's the probability of selecting a vanilla ice cream given that a chocolate one hasn't been chosen?

Solution:

• Total probability excluding chocolate:

• Probability of vanilla under the given condition:

Problem 2

In a survey, 60 out of 100 people own a car, 40 own a bike, and 15 own both. What's the probability that a person owns a bike given they own a car?

Solution:

• Joint probability of owning both a car and a bike:

• Probability of owning a car:

• Calculating the conditional probability: