🎲 Probability – Core Concepts
Events, probability and probability trees
🎯 What Is Probability?
Probability is a number between 0 and 1 that describes the likelihood that something will happen.
| 0 Impossible Will never happen | 0.5 Equal chance 50-50 | 1 Certain Will definitely happen |
📚 Core Concepts
| Concept | Definition | Example |
|---|---|---|
| Experiment | An action whose outcome is not known in advance | Rolling a die, tossing a coin |
| Sample space (Ω) | The set of all possible outcomes | Die: \(\{1,2,3,4,5,6\}\) |
| Event | A subset of the sample space | "Even number": \(\{2,4,6\}\) |
| Simple event | An event with exactly one outcome | "Got 3": \(\{3\}\) |
⭐ Basic Probability Formula
\(P(A) = \frac{\text{number of favourable outcomes}}{\text{number of possible outcomes}}\)
Example: Rolling a fair die
Question: What is the probability of rolling an even number?
Sample space: \(\{1, 2, 3, 4, 5, 6\}\) → 6 possible outcomes
Event "even number": \(\{2, 4, 6\}\) → 3 favourable outcomes
\(P(\text{even}) = \frac{3}{6} = \frac{1}{2}\)
⚠️ Important: this formula works only when all outcomes are equally likely!
🔄 Complementary Event
The complement of event \(A\) (denoted \(\bar{A}\) or \(A'\)) is:
"The event did not occur"
\(P(A) + P(\bar{A}) = 1\)
Therefore: \(P(\bar{A}) = 1 - P(A)\)
Example:
If \(P(\text{rain}) = 0.3\)
then \(P(\text{no rain}) = 1 - 0.3 = 0.7\)
💡 When to use it?
When it is easier to calculate the probability that something will not happen.
For example: "at least one" → easier to calculate "none at all" and subtract from 1.
🔗 Operations on Events
| Operation | Notation | Meaning | Key word |
|---|---|---|---|
| Union | \(A \cup B\) | \(A\) occurred or \(B\) occurred (or both) | "or" |
| Intersection | \(A \cap B\) | \(A\) occurred and \(B\) occurred | "and" |
| Complement | \(\bar{A}\) | \(A\) did not occur | "not" |
⊘ Mutually Exclusive Events
Two events are mutually exclusive if they cannot both occur at the same time
\(A \cap B = \emptyset\)
If \(A\) and \(B\) are mutually exclusive:
\(P(A \cup B) = P(A) + P(B)\)
(simply add them!)
Example:
When rolling a die: "got 1" and "got 6" are mutually exclusive.
(you cannot get both 1 and 6 on the same roll)
\(P(1 \text{ or } 6) = P(1) + P(6) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}\)
🌳 Probability Tree
A probability tree is a visual tool for describing an experiment with multiple stages.
Each branch represents an outcome, and the probability is written beside each branch.
Example: Tossing a coin twice
H = Heads, T = Tails
📐 Tree Calculation Rules
| Rule | When to use | Example |
|---|---|---|
| Multiplication rule Along a path Multiply the probabilities | To calculate probability of a specific path | \(P(\text{H,H}) = 0.5 \times 0.5 = 0.25\) |
| Addition rule Between paths Add the probabilities | To calculate probability of multiple paths (or) | \(P(\text{exactly one H})\) \(= P(\text{H,T}) + P(\text{T,H})\) \(= 0.25 + 0.25 = 0.5\) |
💡 How to remember?
"and" = multiply (along a path – this and this and this…)
"or" = add (between paths – either this or that)
✏️ Detailed Example
Question: A bag contains 3 red balls and 2 blue balls. A ball is drawn, replaced, then drawn again. Find the probability of:
a. Two red balls?
b. Exactly one red ball?
c. At least one red ball?
Given:
\(P(\text{red}) = \frac{3}{5}\) \(P(\text{blue}) = \frac{2}{5}\)
a. Two red balls (R,R):
\(P(\text{R,R}) = \frac{3}{5} \times \frac{3}{5} = \frac{9}{25}\)
b. Exactly one red ball (R,B or B,R):
\(P(\text{R,B}) = \frac{3}{5} \times \frac{2}{5} = \frac{6}{25}\)
\(P(\text{B,R}) = \frac{2}{5} \times \frac{3}{5} = \frac{6}{25}\)
\(P(\text{exactly one red}) = \frac{6}{25} + \frac{6}{25} = \frac{12}{25}\)
c. At least one red ball:
Use the complement! "at least one" = 1 − "none at all"
\(P(\text{B,B}) = \frac{2}{5} \times \frac{2}{5} = \frac{4}{25}\)
\(P(\text{at least one red}) = 1 - \frac{4}{25} = \frac{21}{25}\)
✅ Self-Check
The sum of all probabilities in a tree must equal 1!
From the previous example:
\(P(\text{R,R}) + P(\text{R,B}) + P(\text{B,R}) + P(\text{B,B})\)
\(= \frac{9}{25} + \frac{6}{25} + \frac{6}{25} + \frac{4}{25} = \frac{25}{25} = 1\) ✓
💡 Tip: If the sum is not 1, there is a calculation error!
💡 Important Tips for the Exam
1️⃣ Always draw a tree!
For multi-stage questions – a tree helps you see all possibilities and not miss any paths
2️⃣ "At least" = complement
"At least one" is hard to calculate directly.
Better: \(1 - P(\text{none})\)
3️⃣ Check logic
Probability is always between 0 and 1!
Got more than 1? There is an error.
4️⃣ With/without replacement
With replacement: probabilities do not change
Without replacement: probabilities change!
📝 Summary
\(P(A) = \frac{\text{favourable}}{\text{possible}}\) | \(P(\bar{A}) = 1 - P(A)\)
In a tree: along a path = multiply | between paths = add
Mutually exclusive events: \(P(A \cup B) = P(A) + P(B)\)