Venn diagrams and tree diagrams

Venn diagrams

A Venn diagram shows events as overlapping circles inside a rectangle (all possible outcomes).
The overlap (intersection) is " $A$ and $B$ "; everything inside either circle is " $A$ or $B$ " (the union).
The region outside both circles is "neither" — the complement of $A$ or $B$ .
Venn diagrams make the addition rule visible: adding the two circles double-counts the overlap, so you subtract it once.

A tree diagram shows events happening in stages, with each branch labelled by its probability.
Multiply along a path (an "and"); add separate paths that both succeed (an "or").
Trees are ideal for conditional situations and without-replacement problems, where the second-stage probabilities depend on the first.

Worked ExampleWithout replacement on a tree

A bag has 5 red and 3 blue counters. Two are drawn without replacement. Find the probability both are red.

$P(\text{red}) = \frac{5}{8}$

Now 4 red remain out of 7 counters:

$P(\text{red} \mid \text{first red}) = \frac{4}{7}$

$P(\text{both red}) = \frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A fair coin is tossed twice. Use a tree diagram idea to find the probability of two tails.

Merit

In a group of 40, 25 like tea, 18 like coffee, and 10 like both. How many like neither? (Use a Venn diagram.)

Excellence

A box has 6 good and 4 faulty bulbs. Two are taken without replacement. Find the probability that exactly one is faulty.

A Venn diagram shows events as overlapping circles inside a rectangle (all possible outcomes).
The overlap (intersection) is " $A$ and $B$ "; everything inside either circle is " $A$ or $B$ " (the union).
The region outside both circles is "neither" — the complement of $A$ or $B$ .
Venn diagrams make the addition rule visible: adding the two circles double-counts the overlap, so you subtract it once.

A tree diagram shows events happening in stages, with each branch labelled by its probability.
Multiply along a path (an "and"); add separate paths that both succeed (an "or").
Trees are ideal for conditional situations and without-replacement problems, where the second-stage probabilities depend on the first.