Conditional Probability

What Is Conditional Probability?

Conditional probability is all about narrowing things down.

It asks: what’s the chance of something happening if we already know something else has happened?

We write this as: $P(A\,|\,B)$ Read as: “the probability of A given B.”

 
Key Ideas

• Conditional probability only considers a smaller part of the sample space — not all the possible outcomes.
• You must restrict your focus to the condition you're given.
• It often appears in problems involving tables, Venn diagrams, or tree diagrams.

Example (Not Conditional):

A spinner has 9 numbers: 1 to 9.

• What’s the probability it lands on a multiple of 3?

Multiples of 3: 3, 6, 9 $\to$ 3 outcomes

Total outcomes: 9

$P(\text{multiple of 3}) = \frac{3}{9} = \frac{1}{3}$

Example (Conditional):

Suppose the spinner only lands on even numbers: 2, 4, 6, 8

• What’s the probability it lands on a multiple of 3 given it landed on an even number?

Only one of the even numbers is a multiple of 3: 6

Total even numbers: 4 $\to$ (2, 4, 6, 8)

$P(\text{multiple of 3} \,|\,\text{even}) = \frac{1}{4}$

Worked Example

A bag contains 80 marbles. Some are red (R), and some are green (G).

 
Find: $P(\text{Shiny} \,|\,\text{Red})$

Step-by-Step:

• We’re told the marble is red, so we only care about the 40 red marbles.
• Out of these 40, 24 are shiny.

$P(\text{Shiny} \,|\,\text{Red}) = \frac{24}{40} = \frac{3}{5}$

Final Answer: $\frac{3}{5}$

 
Example

A group of 50 students were surveyed about their favourite subject.

What is the probability that a student chosen at random prefers Maths, given that the student is a girl?

Step-by-Step:

We’re told the student is a girl: 25 girls in total.

Of these, 12 prefer Maths.

$P(\text{Maths} \,|\,\text{Girl}) = \frac{12}{25}$

Final Answer: $\frac{12}{25}$

Example

Question:

Given the information in the Venn diagram, find the probability that a student:

(a) Likes Maths given that they like Science

(b) Likes neither subject.

(a) P(Maths | Science)

We're told the student likes Science, so the total is only those in set $S$:

$$P(\text{Maths} \mid \text{Science}) = \frac{\text{Number who like both}}{\text{Number who like Science}} = \frac{25}{55} = \frac{5}{11}$$

(b) P(Neither Maths nor Science)

We already calculated this as:

$$P(\text{Neither}) = \frac{30}{100} = 0.3$$