Set Notation And Probability Diagrams

Set Notation Basics

• A set is a collection of elements, e.g. {1, 2, 3, 6} is the set of factors of 6.
• ℰ is the universal set (everything we consider).
• n(A) means the number of elements in set A.
• A ∩ B is the intersection (elements in both A and B).
• A ∪ B is the union (elements in A or B or both).
• A' is the complement (elements not in A).

If ℰ $= {1,2,3,4,5,6}, A = {2,4,6}, B = {1,2,3}$, what is A ∩ B?

• {2}
• {1,2,3,4,6}
• {1,3,5}
• {2,4,6}

Venn Diagrams

• A Venn diagram uses a rectangle for ℰ and circles for sets.
• Overlapping circles show intersections.
• Each region represents a combination of sets.

In a Venn diagram, the region where circles overlap represents the ____ of the sets.

Probabilities from Venn Diagrams

• If the Venn diagram shows frequencies, add frequencies in the desired region and divide by total frequency.
• If it shows elements, count elements in the region and divide by total elements.
• For conditional probability (e.g., given you are in A), use the number in A as the denominator.

In a Venn diagram with total 30, 12 in A only, 8 in B only, 5 in both, and 5 outside. What is P(A ∩ B)?

• $\frac{5}{30}$
• $\frac{12}{30}$
• $\frac{8}{30}$
• $\frac{5}{25}$

Two-Way Tables

• Two-way tables compare two characteristics, e.g., year group and language.
• Add row and column totals, and an overall total in the bottom-right.
• Probabilities are found by dividing the relevant cell (or total) by the appropriate total.

In a two-way table, the overall total should be the sum of the row totals or the sum of the ____ totals.

Probability Tree Diagrams

• Tree diagrams show outcomes of repeated experiments.
• Probabilities on branches from the same point add to 1.
• Multiply along branches to find probability of a sequence.
• Add probabilities of different sequences for 'or'.

A tree diagram shows first event: $P(G)=\frac{5}{7}, P(R)=\frac{2}{7}$. Second event: $P(G)=\frac{8}{9}, P(R)=\frac{1}{9}$. What is P(G then R)?

• $\frac{5}{7} \times \frac{1}{9}$
• $\frac{5}{7} \times \frac{8}{9}$
• $\frac{2}{7} \times \frac{1}{9}$
• $\frac{2}{7} \times \frac{8}{9}$

Combined Probability

• Use 'and' rule: P(A and $B) = P(A) \times P(B)$ for independent events.
• Use 'or' rule: P(A or $B) = P(A) + P(B)$ for mutually exclusive events.
• For 'at least one', use $1 - P(none)$.

Which of these are true about probability?

• P(A and $B) = P(A) \times P(B)$ always.
• $P(not A) = 1 - P(A)$.
• For independent events, P(A and $B) = P(A) \times P(B)$.
• P(A or $B) = P(A) + P(B)$ always.

Worked Example: Venn Diagram

• In a class of 30, 15 study Spanish, 3 study both Spanish and German, 7 study neither.
• Start with the intersection: 3 in both.
• Spanish only $= 15 - 3 = 12$. German only $= 30 - 12 - 3 - 7 = 8$.
• P(Spanish but not $German) = \frac{12}{30} = \frac{2}{5}$.

In the example, the number of students who study German only is ____.

Worked Example: Tree Diagram

• Two sets of traffic lights: $P(first green)=\frac{5}{7}, P(second green)=\frac{8}{9}$.
• Draw branches: first G or R, then second G or R.
• $P(both red) = (\frac{2}{7}) \times (\frac{1}{9}) = \frac{2}{63}$.
• P(at least one $red) = 1 - P(both green) = 1 - (\frac{5}{7} \times \frac{8}{9}) = \frac{23}{63}$.

Using the tree diagram example, what is P(first red and second green)?

• $\frac{2}{7} \times \frac{8}{9}$
• $\frac{2}{7} \times \frac{1}{9}$
• $\frac{5}{7} \times \frac{8}{9}$
• $\frac{5}{7} \times \frac{1}{9}$

Worked Example: Combined Probability

• Box with 3 blue and 8 red counters, with replacement.
• $P(both red) = (\frac{8}{11}) \times (\frac{8}{11}) = \frac{64}{121}$.
• $P(different colours) = P(blue$ then $red) + P(red$ then $blue) = (\frac{3}{11} \times \frac{8}{11}) + (\frac{8}{11} \times \frac{3}{11}) = \frac{48}{121}$.

Which statements about the counter example are true?

• $P(both blue) = (\frac{3}{11})^{2}$
• $P(same colour) = (\frac{8}{11})^{2} + (\frac{3}{11})^{2}$
• $P(different colours) = 1 - P(same colour)$
• The events are independent because counters are replaced.