Probability Distributions
Complete Notes & Formulae for Eleventh Grade (Algebra 2)
1. Identify Probability Distributions
What is a Probability Distribution?
A probability distribution shows all possible values of a random variable and their associated probabilities
Requirements for Valid Probability Distribution:
• Each probability must be between 0 and 1: \( 0 \leq P(X) \leq 1 \)
• Sum of all probabilities equals 1: \( \sum P(X) = 1 \)
• All possible outcomes are included
Example:
Is this a valid probability distribution?
| X | P(X) |
|---|---|
| 1 | 0.2 |
| 2 | 0.3 |
| 3 | 0.5 |
Check: 0.2 + 0.3 + 0.5 = 1.0 ✓
YES, this is valid!
2. Identify Discrete and Continuous Random Variables
Random Variable:
A variable whose value is determined by the outcome of a random event
Discrete Random Variable:
Definition:
Takes on a countable number of distinct values (can count them)
Key Feature: COUNTING
Can list all possible values: 0, 1, 2, 3, ...
Examples:
• Number of students in a class (20, 21, 22...)
• Number of heads when flipping 5 coins (0, 1, 2, 3, 4, 5)
• Number of cars sold per day (0, 1, 2, 3...)
• Roll of a die (1, 2, 3, 4, 5, 6)
Continuous Random Variable:
Definition:
Takes on an infinite number of values within an interval (can measure them)
Key Feature: MEASURING
Cannot list all values; can take any value in a range
Examples:
• Height of students (5.4 ft, 5.47 ft, 5.479 ft...)
• Time to complete a race (measured precisely)
• Weight of an object
• Temperature
3. Write a Discrete Probability Distribution
Steps to Create:
1. List all possible values of the random variable X
2. Find the probability for each value
3. Create a table showing X and P(X)
4. Verify that all probabilities sum to 1
Example:
Flip two fair coins. Let X = number of heads. Write the probability distribution.
Possible outcomes: HH, HT, TH, TT
X can be: 0, 1, or 2
P(X=0) = 1/4 (TT)
P(X=1) = 2/4 = 1/2 (HT, TH)
P(X=2) = 1/4 (HH)
| X | P(X) |
|---|---|
| 0 | 0.25 |
| 1 | 0.50 |
| 2 | 0.25 |
4. Graph a Discrete Probability Distribution
How to Graph:
Create a probability histogram or bar graph:
• X-axis: Values of the random variable
• Y-axis: Probability P(X)
• Draw bars with height equal to probability
• Bars should NOT touch (discrete values)
Key Features:
• Total area of all bars = 1
• Each bar represents P(X = x)
• Higher bars = more likely outcomes
5. Expected Values of Random Variables
Expected Value (Mean):
The expected value (or mean) is the long-run average value of the random variable
\[ E(X) = \mu = \sum [x \cdot P(x)] \]
Formula in words:
Multiply each value by its probability, then add all products
Example:
Find the expected value:
| X | P(X) |
|---|---|
| 1 | 0.2 |
| 2 | 0.5 |
| 3 | 0.3 |
\( E(X) = (1)(0.2) + (2)(0.5) + (3)(0.3) \)
\( E(X) = 0.2 + 1.0 + 0.9 = 2.1 \)
Expected Value: 2.1
6. Variance of Random Variables
Variance Formula:
Variance measures the spread of the distribution (how far values are from the mean)
\[ \text{Var}(X) = \sigma^2 = \sum [(x - \mu)^2 \cdot P(x)] \]
or equivalently
\[ \sigma^2 = \sum [x^2 \cdot P(x)] - \mu^2 \]
Steps:
1. Find the expected value \( \mu \)
2. Subtract \( \mu \) from each value and square the result
3. Multiply each squared difference by its probability
4. Sum all the products
7. Standard Deviation of Random Variables
Standard Deviation Formula:
Standard deviation is the square root of variance (measures typical deviation from mean)
\[ \sigma = \sqrt{\text{Var}(X)} = \sqrt{\sigma^2} \]
Standard deviation has the same units as the original data
Complete Example:
Using the distribution from Section 5:
| X | P(X) | (X - μ)² | (X - μ)² · P(X) |
|---|---|---|---|
| 1 | 0.2 | (1-2.1)² = 1.21 | 0.242 |
| 2 | 0.5 | (2-2.1)² = 0.01 | 0.005 |
| 3 | 0.3 | (3-2.1)² = 0.81 | 0.243 |
Variance: \( \sigma^2 = 0.242 + 0.005 + 0.243 = 0.49 \)
Standard Deviation: \( \sigma = \sqrt{0.49} = 0.7 \)
8. Write Probability Distribution for a Game of Chance
Steps:
1. Identify all possible outcomes (winnings or losses)
2. Calculate probability of each outcome
3. Account for cost to play (subtract from winnings)
4. Create probability distribution table
Example:
A lottery costs $1 to play. You win $100 with probability 0.01, $10 with probability 0.05, and nothing otherwise.
Let X = net profit (winnings - cost)
| Outcome | Net Profit X | P(X) |
|---|---|---|
| Win $100 | $99 | 0.01 |
| Win $10 | $9 | 0.05 |
| Win Nothing | -$1 | 0.94 |
9. Expected Values for a Game of Chance
Interpretation:
Expected value in a game tells you the average profit/loss per play over many games
Interpretation Guidelines:
• If E(X) > 0: Game favors the player (you expect to win)
• If E(X) = 0: Fair game (break even in long run)
• If E(X) < 0: Game favors the house (you expect to lose)
Example (continued from Section 8):
\( E(X) = (99)(0.01) + (9)(0.05) + (-1)(0.94) \)
\( E(X) = 0.99 + 0.45 - 0.94 = 0.50 \)
Expected Value: $0.50
Interpretation: On average, you expect to win $0.50 per game if you play many times
10. Choose the Better Bet
Decision Rule:
Compare expected values to choose the better bet:
1. Calculate E(X) for each game/bet
2. Choose the game with the HIGHER expected value
3. The higher E(X) means better long-term outcome
Example:
Which is the better bet?
Game A:
Pay $5, win $50 with probability 0.1, win $0 with probability 0.9
\( E(A) = (45)(0.1) + (-5)(0.9) = 4.5 - 4.5 = \$0 \)
Game B:
Pay $2, win $20 with probability 0.15, win $0 with probability 0.85
\( E(B) = (18)(0.15) + (-2)(0.85) = 2.7 - 1.7 = \$1 \)
Game B is the better bet because E(B) = $1 > E(A) = $0
11. Quick Reference Summary
Key Formulas:
Expected Value (Mean):
\[ E(X) = \mu = \sum [x \cdot P(x)] \]
Variance:
\[ \sigma^2 = \sum [(x - \mu)^2 \cdot P(x)] \]
Standard Deviation:
\[ \sigma = \sqrt{\sigma^2} \]
| Concept | Key Point |
|---|---|
| Discrete RV | Countable values (counting) |
| Continuous RV | Infinite values in interval (measuring) |
| Valid Distribution | All P(x) between 0 and 1, sum = 1 |
| Expected Value | Long-run average (μ) |
| Variance | Measure of spread (σ²) |
| Better Bet | Higher expected value |
📚 Study Tips
✓ Check that all probabilities sum to 1 for a valid distribution
✓ Expected value is what you expect "on average" over many trials
✓ Standard deviation measures typical distance from the mean
✓ For games of chance, include the cost to play in your calculations
✓ Always choose the bet with the higher expected value