Year 12 Probability: Conditional probability

Conditional probability asks for the probability of one event given that another event has already happened. That condition changes the sample space.

Once a condition is given, stop using the original total blindly. Restrict attention to the outcomes that satisfy the condition first.

Main formula

$P (A | B) = P (A \cap B) / P (B)$ , provided $P (B) \neq 0$ .

Worked example

Problem: In a school, 75 students play a sport, 30 play an instrument and 18 do both. If one student is chosen from those who play a sport, what is the probability that the student also plays an instrument?

The conditioning event is "plays a sport", so the denominator is 75.
The students who satisfy both conditions are the 18 who play a sport and an instrument.
Use the formula: $P (Instrument | Sport) = P (Instrument \cap Sport) / P (Sport)$ .
With counts, this becomes $18 / 75 = 6 / 25$ .

Answer: the conditional probability is

6 / 25

What to check

Identify the conditioning event clearly.
Rewrite the sample space if needed.
Use Venn diagrams or tables when the wording is dense.
Decide whether independence is actually given or only assumed.

Worked example 2

Problem: A bag contains 3 red balls and 5 blue balls. Two balls are drawn without replacement. What is the probability that the second ball is red given that the first ball was blue?

The condition tells us the first ball is already known to be blue.
That leaves 7 balls: 3 red and 4 blue.
The conditional sample space is now those 7 remaining balls.
So the probability that the second ball is red is $3 / 7$ .

Answer: the conditional probability is

3 / 7

Common mistakes

Using the original sample space after the condition has changed it.
Mixing up $P (A | B)$ and $P (B | A)$ .
Assuming events are independent when the question gives no reason to do so.

Revision focus

Probability work becomes far more reliable when you define the object of the question before calculating. Is the problem about ordered outcomes, unordered selections, a conditional event, a discrete random variable, or a continuous density? That decision controls the method. In many exam questions, the major error is not arithmetic. It is using a correct formula for the wrong kind of probability situation.

A second useful habit is to interpret the result immediately after it is found. Decide whether the answer is a count, a probability, an average outcome, or a spread measure. In continuous settings, remember that probabilities come from areas over intervals, not from single points. In discrete settings, remember that the total probability must still add to one. Those interpretation checks are often the fastest way to catch an error before it reaches the final line.

In probability, the final line should always answer the right event, not just display a calculation. After computing, read the question again and confirm that your sample space, condition, or random-variable interpretation still matches what was asked. That last comparison catches a surprising number of near-miss errors.