Probability: Events, Venn Diagrams & the Addition Rule

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Probability measures how likely something is to happen. It’s always a number between 0 (impossible) and 1 (certain). In Grade 10, you learn to calculate probabilities using sample spaces, Venn diagrams, and the addition rule.

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The Basics

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$$P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$$

Probability	Meaning
$P = 0$	Impossible
$P = 0.5$	Even chance (50/50)
$P = 1$	Certain

Theoretical vs Relative Frequency

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Type	How you find it	Example
Theoretical	Using logic/counting	$P(\text{heads}) = \frac{1}{2}$
Relative frequency	Using experimental data	Flipped 100 times, got 47 heads → $P = 0.47$

As the number of trials increases, the relative frequency gets closer to the theoretical probability.

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Sample Space & Events

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• Sample space ($S$): The set of ALL possible outcomes
• Event ($E$): A subset of outcomes you’re interested in
• Complement ($E'$ or “not $E$”): Everything in $S$ that is NOT in $E$

$$P(E') = 1 - P(E)$$

Example: Rolling a die. $S = \{1, 2, 3, 4, 5, 6\}$. If $E$ = “rolling even” = $\{2, 4, 6\}$, then $P(E) = \frac{3}{6} = \frac{1}{2}$.

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Venn Diagrams

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A Venn diagram organises events visually:

• Rectangle = sample space (everything)
• Circles = events
• Overlap = both events happening ($A \text{ and } B$)
• Outside circles = neither event

Always fill from the INSIDE out: Start with the overlap, then the “only” regions, then “neither”.

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The Addition Rule

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$$\boxed{P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)}$$

You subtract the overlap to avoid counting it twice.

Mutually Exclusive Events

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If two events cannot happen at the same time, they are mutually exclusive:

$$P(A \text{ and } B) = 0 \quad \Rightarrow \quad P(A \text{ or } B) = P(A) + P(B)$$

Example: Rolling a 3 and rolling a 5 on one die — impossible to get both, so they’re mutually exclusive.

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Deep Dives

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• Probability Basics & Calculations — sample spaces, listing outcomes, calculating probabilities
• Venn Diagrams & the Addition Rule — filling Venn diagrams, mutually exclusive events, the complement rule

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🚨 Common Mistakes

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1. Probability > 1: If your answer is greater than 1 or negative, you’ve made an error. Probability is always between 0 and 1.
2. Forgetting “neither”: In Venn diagrams, always check if there are outcomes outside both circles.
3. Double counting in the addition rule: $P(A \text{ or } B) \neq P(A) + P(B)$ unless they’re mutually exclusive. You must subtract the overlap.
4. “Or” means add (roughly), “And” means multiply (roughly): This becomes more precise in Grade 11.

🔗 Related Grade 10 topics:

• Statistics — data analysis connects to probability

📌 Where this leads in Grade 11: Probability: Independent & Dependent Events — tree diagrams, contingency tables, and the product rule

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