Probability & Venn Diagrams

Basic Probability Concepts

Probability measures how likely an event is to happen. It is always a number between 0 and 1, inclusive:

• Probability 0 means the event is impossible.
• Probability 1 means the event is certain.
• Probabilities between 0 and 1 indicate how likely an event is, with values closer to 1 meaning more likely.

The probability scale can be visualised as:

Impossible (0) ———— Unlikely ———— Even chance (0.5) ———— Likely ———— Certain (1)

Probability is calculated by:

$$\text{Probability of event} = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$$

For instance, when rolling a fair six-sided die, the probability of rolling a 4 is:

$$\frac{1}{6} \approx 0.167$$

Venn Diagrams Basics

Venn diagrams are visual tools used to represent sets and their relationships. Each set is shown as a circle, and elements belonging to sets are placed inside the circles.

Key terms:

• Set: A collection of elements or outcomes.
• Element: An individual member of a set.
• Union (A 22 B): All elements in set A or set B (or both).
• Intersection (A 23 B): Elements common to both sets A and B.
• Complement (A^c): Elements not in set A.

For example, if set A is students who play football and set B is students who play tennis, the union A 22 B represents all students who play football or tennis or both, while the intersection A 23 B represents students who play both sports.

Probability with Venn Diagrams

Venn diagrams can be used to calculate probabilities of combined events by representing events as sets.

Mutually exclusive events are events that cannot happen at the same time. Their intersection is empty, so:

$$P(A \cap B) = 0$$

For mutually exclusive events, the probability of either event A or event B occurring is:

$$P(A \cup B) = P(A) + P(B)$$

Non-mutually exclusive events can happen at the same time, so their intersection is not empty. The general addition rule applies:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

This subtracts the overlap once to avoid double counting.

Set notation is used in probability to describe events clearly:

• $P(A)$: Probability of event A
• $P(B)$: Probability of event B
• $P(A \cap B)$: Probability of both A and B occurring
• $P(A \cup B)$: Probability of A or B (or both) occurring
• $P(A^c)$: Probability of event A not occurring (complement)

Using a Venn diagram helps visualise these probabilities by shading the relevant regions.

For example, if $P(A) = 0.5$, $P(B) = 0.4$, and $P(A \cap B) = 0.1$, then:

$$P(A \cup B) = 0.5 + 0.4 - 0.1 = 0.8$$

Multiple Events and Combined Probability

When dealing with multiple events, the addition rule helps find the probability of one or more events occurring.

For two events A and B:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

If events are mutually exclusive, $P(A \cap B) = 0$, so the formula simplifies to:

$$P(A \cup B) = P(A) + P(B)$$

Venn diagrams are useful to represent overlapping events and calculate combined probabilities by shading the union area.

For example, if two events overlap, the intersection probability must be subtracted once to avoid counting it twice.

If you know the probabilities of individual events and their intersection, you can find the probability of either event occurring.

For three or more events, the addition rule extends but is more complex (covered in other topics). For more details, see the topic on Combined Probability for multiple events.

Example: Suppose in a class, 70\% of students like chocolate (C), 50\% like vanilla (V), and 30\% like both. Find the probability a student likes chocolate or vanilla.

Using the addition rule:

$$P(C \cup V) = P(C) + P(V) - P(C \cap V) = 0.7 + 0.5 - 0.3 = 0.9$$