Probability
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Calculate Probabilities of Events
Basic Formula:
\[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]
• Probability ranges from 0 to 1 (or 0% to 100%)
• \( P(A) = 0 \): Event A is impossible
• \( P(A) = 1 \): Event A is certain
• \( 0 < P(A) < 1 \): Event A may or may not occur
Example:
Find probability of rolling a 4 on a standard die
Favorable outcomes: 1 (rolling a 4)
Total outcomes: 6 (faces on die)
\( P(4) = \frac{1}{6} \approx 0.167 \) or 16.7%
2. Permutations
Definition:
A permutation is an arrangement of objects where order matters
\[ P(n, r) = \frac{n!}{(n-r)!} \]
where:
• \( n \) = total number of items
• \( r \) = number of items being arranged
• \( n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1 \)
Example:
How many ways can 3 students be arranged from a class of 5?
\( n = 5 \), \( r = 3 \)
\( P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{120}{2} = 60 \)
60 different arrangements
3. Combinations
Definition:
A combination is a selection of objects where order does not matter
\[ C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where:
• \( n \) = total number of items
• \( r \) = number of items being selected
Example:
How many ways can you choose 3 students from a class of 5?
\( n = 5 \), \( r = 3 \)
\( C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \cdot 2!} = \frac{120}{6 \cdot 2} = \frac{120}{12} = 10 \)
10 different selections
4. Permutations vs Combinations
| Aspect | Permutations | Combinations |
|---|---|---|
| Order | Order matters | Order does not matter |
| Formula | \( P(n,r) = \frac{n!}{(n-r)!} \) | \( C(n,r) = \frac{n!}{r!(n-r)!} \) |
| Example | Password, Race positions | Committee selection, Lottery |
| Result | Larger value | Smaller value |
5. Two-Way Frequency Tables
Definition:
A two-way frequency table displays data for two categorical variables
Example Table:
| Male | Female | Total | |
|---|---|---|---|
| Plays Sports | 30 | 20 | 50 |
| Doesn't Play | 10 | 40 | 50 |
| Total | 40 | 60 | 100 |
Probability from table:
P(Female) = \( \frac{60}{100} = 0.6 \)
P(Male and Plays Sports) = \( \frac{30}{100} = 0.3 \)
6. Independent Events
Definition:
Two events are independent if the occurrence of one does not affect the probability of the other
\[ P(A \text{ and } B) = P(A) \times P(B) \]
\[ \text{Or: } P(A|B) = P(A) \]
Example:
Flipping a coin twice - are the events independent?
Event A: First flip is heads, P(A) = 0.5
Event B: Second flip is heads, P(B) = 0.5
P(A and B) = 0.5 × 0.5 = 0.25
Yes, they are independent events!
7. Conditional Probability
Definition:
Conditional probability is the probability of event A occurring given that event B has already occurred
\[ P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \]
Read as: "Probability of A given B"
Example:
Using the sports table, find P(Plays Sports | Female)
P(Plays Sports and Female) = \( \frac{20}{100} = 0.2 \)
P(Female) = \( \frac{60}{100} = 0.6 \)
P(Plays Sports | Female) = \( \frac{0.2}{0.6} = \frac{20}{60} = \frac{1}{3} \)
≈ 0.333 or 33.3%
8. Addition Rule for Probability
General Addition Rule:
For any two events A and B:
\[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \]
Mutually Exclusive Events:
If events cannot happen at the same time (mutually exclusive), then P(A and B) = 0:
\[ P(A \text{ or } B) = P(A) + P(B) \]
Examples:
P(rolling a 3 or 5 on a die)
These are mutually exclusive events
P(3) = \( \frac{1}{6} \), P(5) = \( \frac{1}{6} \)
P(3 or 5) = \( \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \)
Drawing a King or a Heart from a deck
Not mutually exclusive (King of Hearts exists)
P(King) = \( \frac{4}{52} \), P(Heart) = \( \frac{13}{52} \), P(King and Heart) = \( \frac{1}{52} \)
P(King or Heart) = \( \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13} \)
9. Quick Reference Summary
Key Formulas:
Basic Probability: \( P(A) = \frac{\text{favorable}}{\text{total}} \)
Permutations: \( P(n,r) = \frac{n!}{(n-r)!} \) (order matters)
Combinations: \( C(n,r) = \frac{n!}{r!(n-r)!} \) (order doesn't matter)
Independent Events: \( P(A \text{ and } B) = P(A) \times P(B) \)
Conditional Probability: \( P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \)
Addition Rule: \( P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \)
Mutually Exclusive: \( P(A \text{ or } B) = P(A) + P(B) \)
📚 Study Tips
✓ Probability always between 0 and 1
✓ Permutations when order matters, combinations when it doesn't
✓ Independent events: one doesn't affect the other
✓ Conditional probability: given that something already happened
✓ Use addition rule for "or" probabilities (subtract overlap if not mutually exclusive)