AP Precalculus: Probability Distributions

Master expected value, variance, and discrete probability distributions

📊 Distributions 📈 Expected Value 📉 Variance 🎲 Games of Chance

📚 Understanding Probability Distributions

A probability distribution describes all possible values of a random variable and their associated probabilities. It provides a complete picture of the likelihood of each outcome. Understanding distributions helps us calculate expected values, measure variability, and make informed decisions in uncertain situations.

1 Discrete vs. Continuous Random Variables

A random variable assigns numerical values to outcomes of a random process. It can be discrete or continuous.

Discrete Random Variable

Takes countable values (finite or countably infinite). You can list all possible outcomes.

Examples: Dice roll, number of students, coin flips

Continuous Random Variable

Takes any value within an interval. Infinite possibilities in any range.

Examples: Height, time, temperature, distance

💡 Quick Test

If you can count the outcomes (1, 2, 3...) → Discrete. If you measure the outcome on a continuous scale → Continuous.

2 Discrete Probability Distribution

A discrete probability distribution lists each value of the random variable with its probability. The probabilities must satisfy two conditions.

Condition 1

$0 \leq P(x_i) \leq 1$

Condition 2

$\sum P(x_i) = 1$

Example Distribution Table

Outcome $x$	0	1	2	3
$P(X = x)$	0.1	0.3	0.4	0.2
Sum	$0.1 + 0.3 + 0.4 + 0.2 = 1.0$ ✓

Probability Mass Function (PMF)

$P(X = x)$ gives the probability that X equals exactly x

Probability Histogram

Bar graph where bar height = probability for each outcome

3 Expected Value (Mean)

The expected value E[X] (or μ) is the long-run average value of the random variable. It's the "weighted average" where outcomes are weighted by their probabilities.

Expected Value Formula $\mu = E[X] = \sum x_i \cdot P(x_i)$

Multiply each value by its probability, then add all products

📌 Example

Using the distribution from above:

$E[X] = 0(0.1) + 1(0.3) + 2(0.4) + 3(0.2)$

$= 0 + 0.3 + 0.8 + 0.6 = 1.7$

Interpretation: On average, over many trials, X will be about 1.7

💡 Key Insight

Expected value doesn't have to be a possible outcome. It represents the center of the distribution, not an actual result.

4 Variance & Standard Deviation

Variance measures how spread out the values are from the mean. Standard deviation is the square root of variance, in the same units as the data.

Variance Formula $\sigma^2 = \text{Var}(X) = \sum (x_i - \mu)^2 \cdot P(x_i)$

Standard Deviation $\sigma = \sqrt{\text{Var}(X)}$

Alternative Variance Formula (Shortcut)

Computational Formula $\sigma^2 = E[X^2] - (E[X])^2 = \sum x_i^2 \cdot P(x_i) - \mu^2$

📌 Example (Using shortcut)

Given: $E[X] = 1.7$ from previous example

Find $E[X^2]$: $0^2(0.1) + 1^2(0.3) + 2^2(0.4) + 3^2(0.2) = 0 + 0.3 + 1.6 + 1.8 = 3.7$

Variance: $\sigma^2 = 3.7 - (1.7)^2 = 3.7 - 2.89 = 0.81$

Standard Deviation: $\sigma = \sqrt{0.81} = 0.9$

5 Games of Chance & Decision Making

Expected value helps determine the best option in games or decisions involving chance. The option with the highest expected value is the best long-term choice.

Expected Value for Games $E = \sum (\text{gain or loss}) \times P(\text{that outcome})$

Fair Game

$E = 0$ — Neither player has an advantage

Favorable Game

$E > 0$ — You expect to win money over time

Unfavorable Game

$E < 0$ — You expect to lose money over time

📌 Example: Is This a Fair Game?

Game: Pay \$5 to play. Roll a die: win \$10 if 6, win \$0 otherwise.

Outcomes: Net gain of \$5 (with P = 1/6) or net loss of \$5 (with P = 5/6)

Expected value: $E = 5(\frac{1}{6}) + (-5)(\frac{5}{6}) = \frac{5}{6} - \frac{25}{6} = -\frac{20}{6} \approx -\$3.33$

Conclusion: Unfair! On average, you lose \$3.33 per game.

⚠️ Casino Games

Almost all casino games have negative expected value for players. The "house edge" ensures the casino profits in the long run.

6 Creating Probability Distributions

To create a probability distribution, list all outcomes, assign probabilities, and verify they sum to 1.

Step 1: Identify all possible values of the random variable
Step 2: Calculate or assign probability for each value
Step 3: Verify that all probabilities are between 0 and 1
Step 4: Verify that probabilities sum to exactly 1
Step 5: Create table or histogram for visualization

📌 Example: Two Coin Flips

Random variable: X = number of heads

Sample space: {HH, HT, TH, TT}

Distribution:

• P(X = 0) = 1/4 (TT)

• P(X = 1) = 2/4 = 1/2 (HT, TH)

• P(X = 2) = 1/4 (HH)

Check: 1/4 + 1/2 + 1/4 = 1 ✓

7 Properties & Rules

These properties help simplify calculations with expected values and variances.

Expected Value Properties

$E[c] = c$ for any constant c
$E[cX] = c \cdot E[X]$
$E[X + Y] = E[X] + E[Y]$
$E[aX + b] = a \cdot E[X] + b$

Variance Properties

$\text{Var}(c) = 0$ for any constant c
$\text{Var}(cX) = c^2 \cdot \text{Var}(X)$
$\text{Var}(aX + b) = a^2 \cdot \text{Var}(X)$

📋 Quick Reference

Expected Value

$\mu = \sum x_i \cdot P(x_i)$

Variance

$\sigma^2 = \sum (x_i - \mu)^2 P(x_i)$

Variance (Shortcut)

$\sigma^2 = E[X^2] - (E[X])^2$

Standard Deviation

$\sigma = \sqrt{\sigma^2}$

Valid Distribution

$\sum P(x_i) = 1$

Fair Game

$E = 0$

Outcome \(x\)	0	1	2	3
\(P(X = x)\)	0.1	0.3	0.4	0.2
Sum	\(0.1 + 0.3 + 0.4 + 0.2 = 1.0\) ✓

📚 Understanding Probability Distributions

1 Discrete vs. Continuous Random Variables

2 Discrete Probability Distribution

Example Distribution Table

Probability Mass Function (PMF)

Probability Histogram

3 Expected Value (Mean)

4 Variance & Standard Deviation

Alternative Variance Formula (Shortcut)

5 Games of Chance & Decision Making

Fair Game

Favorable Game

Unfavorable Game

6 Creating Probability Distributions

7 Properties & Rules

Expected Value Properties

Variance Properties

📋 Quick Reference

Expected Value

Variance

Variance (Shortcut)

Standard Deviation

Valid Distribution

Fair Game

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