Venn diagrams, tree diagrams and conditional probability

Reading a Venn diagram

A Venn diagram shows students who play Football (F) and Basketball (B). 8 play both, 12 play only Football, 5 play only Basketball. How many students total?

Regions in a Venn diagram don't overlap (except at intersections). The union includes all regions.

Drawing a tree diagram for independent events

A coin is flipped twice. Show all outcomes and their probabilities.

For independent events, multiply probabilities along a path. For two flips, each of 4 outcomes has probability 0.5 × 0.5 = 0.25.

Conditional probability without replacement

A bag has 5 red and 3 blue balls. Draw two without replacement. Find P(both red).

Without replacement, the second draw has fewer items. Conditional probability accounts for the changed sample space.

Using conditional probability formula

In a group, 60% have a driving licence (D) and 20% have both a licence and a car (D and C). Find P(car | licence).

Conditional probability restricts the sample space to only the cases where the condition (D) is true.

In a Venn diagram, A has 10 elements, B has 6 elements, and both have 3 in common. How many elements are in A or B?

💡 A only: 10 − 3 = 7. B only: 6 − 3 = 3. Total: 7 + 3 + 3 = 13.

A coin is flipped and a die is rolled. How many outcomes are in the sample space?

💡 2 outcomes for the coin × 6 outcomes for the die = 12 total outcomes.

A bag has 4 red and 6 blue marbles. Draw without replacement. Find P(both red).

💡 P(red first) = $\frac{4}{10}$. P(red second | red first) = $\frac{3}{9}$. Multiply: ($\frac{4}{10}$) × ($\frac{3}{9}$) = $\frac{12}{90}$ = $\frac{2}{15}$.

P(A) = 0.4, P(B) = 0.5, P(A and B) = 0.2. Find P(A | B).

💡 P(A|B) = P(A and B) / P(B) = 0.2 / 0.5 = 0.4.

A tree diagram has first branch probabilities 0.3 and 0.7. What is the sum of all final outcome probabilities?

💡 The sum of all final outcome probabilities must equal 1 (certainty).