Conditional probability Revision Notes

Conditional probability is the probability that one event occurs given that another event has already happened. It is written as P(A|B), which means 'the probability of event A occurring given event B has occurred'. This concept is important because the occurrence of event B can affect the likelihood of event A.

For example, if you know it is raining (event B), the probability that you carry an umbrella (event A) might be higher than if you had no information about the weather. Thus, P(A|B) can be different from the unconditional probability P(A).

• P(A|B) reads as 'probability of A given B'.
• Conditional probability depends on the occurrence of event B.
• It adjusts the sample space to only outcomes where B happens.

Venn diagrams visually represent events as circles within a sample space. The overlap between two circles shows outcomes common to both events.

To find P(A|B) using a Venn diagram, focus on the part of event B's circle. The conditional probability is the proportion of B's area that also lies within A.

This means you divide the number of outcomes in both A and B by the total number of outcomes in B.

• Identify the overlap area between A and B.
• Count or measure outcomes in A ∩ B and in B.
• Calculate P(A|B) = (Number in A ∩ B) ÷ (Number in B).
• Example: If 30 students like football (B), and 18 of these also like basketball (A), then P(A|B) = 18/30 = 0.6.

Tree diagrams show all possible outcomes of sequential events with branches representing each outcome and their probabilities.

To find P(A|B), follow the branches where event B occurs first, then event A occurs after.

Multiply the probabilities along the branches to find joint probabilities, then use these to find conditional probabilities by dividing the joint probability by the probability of B.

• Draw branches for event B outcomes first.
• From each B branch, draw branches for event A outcomes.
• Calculate joint probabilities by multiplying along branches.
• Find P(A|B) = P(A and B) ÷ P(B).
• Example: If the probability of B is 0.5 and the probability of A given B is 0.4, then the joint probability P(A and B) = 0.5 × 0.4 = 0.2.

Tables organise data showing frequencies or probabilities of combined events.

To find P(A|B), look at the frequency of outcomes where both A and B occur and divide by the total frequency of B.

This method is especially useful when dealing with survey data or experimental results.

• Identify the frequency of A and B occurring together.
• Identify the total frequency of B.
• Calculate P(A|B) = Frequency of A and B ÷ Frequency of B.
• Example: In a table, if 12 people like tea and coffee (A and B), and 20 people like coffee (B), then P(A|B) = 12/20 = 0.6.

By using Venn diagrams, tree diagrams, and tables, you can determine conditional probabilities through visual and logical reasoning rather than memorising formulas.

The key is to focus on the reduced sample space where event B has occurred and then find the proportion of that space where event A also occurs.

This approach builds understanding and helps solve a variety of problems involving conditional probability.

• Conditional probability adjusts the sample space to event B.
• Visual tools help identify relevant outcomes.
• Calculations involve dividing the number of favourable outcomes by the total outcomes in B.
• Understanding these methods avoids reliance on formula memorisation.