Probability from Venn Diagrams

Grade 5

From diagram to probability

Once a Venn diagram is filled in, every probability question is just reading the right region and dividing by the total. The tricky part is knowing which region the question is asking about.

Get into the habit of: (1) labelling the total, (2) identifying the region from the notation, (3) writing the fraction.

Worked example — full Venn probability

EXAMPLE

80 students were surveyed about pets. 45 own a cat (C), 30 own a dog (D), 15 own both. Find (a) P(C ∩ D), (b) P(C ∪ D), (c) P(C′ ∩ D′).

Fill in the diagram: Overlap (both): 15. Cat only: 45 − 15 = 30. Dog only: 30 − 15 = 15. Neither: 80 − (30 + 15 + 15) = 20. (a) C ∩ D = overlap = 15 students. P(C ∩ D) = 15/80 = 3/16. (b) C ∪ D = everything in either circle = 30 + 15 + 15 = 60. P(C ∪ D) = 60/80 = 3/4. (c) C′ ∩ D′ = neither cat nor dog = 20. P(C′ ∩ D′) = 20/80 = 1/4.

The neither region

C′ ∩ D′ is the same as (C ∪ D)′ — it's the region outside both circles. Questions often refer to it as "neither" in the wording. In the diagram, it's the part inside the rectangle but outside all the circles.

Adding probabilities from a Venn diagram

An alternative approach: if you have probabilities written in each region (not frequencies), the rules are the same — just add the probability values in the relevant regions instead of counting people.

Whether you're given frequencies or probabilities in the diagram, the method is identical: identify the region, total the values in it, divide by the grand total (or just add them if they're already probabilities that sum to 1 across all regions).

Try this one

For two sets \( A \) and \( B \), \( n(A) = 20 \), \( n(B) = 14 \) and \( n(A \cup B) = 27 \).

(a) Work out \( n(A \cap B) \). [2]

(b) Work out \( n(A \cap B') \), the number in \( A \) only. [2]

Nova's hint:

This uses the addition rule connecting the sizes of two sets, their union and their overlap.

Ask yourself: 'when I add \( n(A) \) and \( n(B) \), how many times is the overlap counted?' Rearranging the rule gives the size of the intersection.

Try to Solve it with Nova?