IB Mathematics AA – Topic 4: Statistics & Probability

Comprehensive Guide to Probability Distributions

Introduction to Probability Distributions

Probability distributions provide mathematical models for random phenomena, describing all possible values a random variable can take and the probabilities associated with each value. From manufacturing quality control and medical testing to financial modeling and weather forecasting, probability distributions enable quantitative predictions about uncertain outcomes.

Key concepts: A random variable assigns numerical values to outcomes of random experiments. Discrete random variables take specific separated values (like number of heads in coin flips), while continuous random variables can take any value in an interval (like heights or temperatures). The binomial distribution models fixed-trial success/failure experiments, while the normal distribution describes many naturally occurring phenomena with its characteristic bell-shaped curve.

Why distributions matter: Rather than treating every problem uniquely, probability distributions provide standard models that apply across countless situations. Understanding that exam scores often follow normal distributions, manufacturing defects follow binomial patterns, and customer arrivals follow Poisson distributions enables professionals to make accurate predictions, set appropriate standards, and quantify risks systematically.

In this guide: We'll master random variables and their probability distributions, understand expected value and variance as measures of central tendency and spread, develop complete proficiency with the binomial distribution for fixed-trial experiments, explore the normal distribution with standardization and z-scores, use tables and GDC to find probabilities, and solve real-world applications—all essential skills for IB exam success.

1. Random Variables and Probability Distributions

What is a Random Variable?

Random Variable Definition:

A random variable (usually denoted by capital letters like \(X\), \(Y\), \(Z\)) is a variable whose value depends on the outcome of a random experiment.

Discrete random variable: Takes specific, separate values (countable)
Continuous random variable: Can take any value in an interval (measurable)

Examples: \(X\) = number of heads in 5 coin flips (discrete), \(Y\) = height of randomly selected person (continuous)

Probability Distribution

Discrete Probability Distribution

A probability distribution shows all possible values and their probabilities

Requirements:

\(0 \leq P(X = x) \leq 1\) for all values
\(\sum P(X = x) = 1\) (sum of all probabilities equals 1)

Expected Value and Variance

Expected Value (Mean): \(E(X)\) or \(\mu\)

\(E(X) = \sum x \cdot P(X = x)\)

Weighted average of all possible values

Variance: \(\text{Var}(X)\) or \(\sigma^2\)

\(\text{Var}(X) = E(X^2) - [E(X)]^2\)

or \(\text{Var}(X) = \sum (x - \mu)^2 \cdot P(X = x)\)

Measures spread of distribution

Standard Deviation: \(\sigma\)

\(\sigma = \sqrt{\text{Var}(X)}\)

Square root of variance; in same units as \(X\)

⚠ Common Pitfalls:

Sum check: Always verify probabilities sum to 1
E(X) vs E(X²): These are different! \(E(X^2) \neq [E(X)]^2\)
Variance formula: Use \(E(X^2) - [E(X)]^2\) to avoid calculation errors
Units: Variance has squared units; standard deviation has original units

2. Binomial Distribution

When to Use Binomial Distribution

Binomial Conditions (BINS):

Binary outcomes: Each trial has only two outcomes (success/failure)
Independent trials: Trials don't affect each other
Number of trials fixed: Know exactly how many trials (\(n\))
Same probability: Probability of success (\(p\)) constant for each trial

Examples: Coin flips, manufacturing defects (with replacement), multiple choice guessing

Binomial Distribution Notation and Formulas

Binomial Distribution \(X \sim B(n, p)\)

Notation: \(X \sim B(n, p)\) means \(X\) follows binomial distribution

\(n\): number of trials
\(p\): probability of success on each trial
\(X\): number of successes in \(n\) trials

Probability Formula:

\(P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}\)

where \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\) is the binomial coefficient

Expected Value and Variance:

\(E(X) = np\)

\(\text{Var}(X) = np(1-p)\)

\(\sigma = \sqrt{np(1-p)}\)

Using GDC for Binomial

Calculator Functions:

binompdf(\(n\), \(p\), \(r\)): Finds \(P(X = r)\) exactly
binomcdf(\(n\), \(p\), \(r\)): Finds \(P(X \leq r)\) cumulative
\(P(X \geq r)\): Calculate as \(1 - P(X \leq r-1)\)
\(P(a \leq X \leq b)\): Calculate as \(P(X \leq b) - P(X \leq a-1)\)

💡 Binomial Tips:

Always use GDC for calculations—faster and more accurate
Check BINS conditions before using binomial distribution
For "at least" questions: use complement or cumulative probability
Mean always equals \(np\)—quick way to check reasonableness

Example 1: Binomial Distribution (IB-Style)

Problem: A fair coin is flipped 10 times. Let \(X\) be the number of heads.

(a) Write down the distribution of \(X\)

(b) Find \(P(X = 6)\)

(d) Find the expected number of heads and standard deviation

Solution:

(a) Distribution:

Check BINS conditions:

✓ Binary: heads or tails

✓ Independent: coin flips don't affect each other

✓ Number fixed: \(n = 10\)

✓ Same probability: \(p = 0.5\) each flip

\(X \sim B(10, 0.5)\)

(b) \(P(X = 6)\):

Using GDC: binompdf(10, 0.5, 6)

\(P(X = 6) = 0.205\)

(c) \(P(X \geq 8)\):

\(P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10)\)

Method 1: Direct calculation

Using GDC: binompdf(10, 0.5, 8) + binompdf(10, 0.5, 9) + binompdf(10, 0.5, 10)

\(= 0.0439 + 0.0098 + 0.00098 = 0.0547\)

Method 2: Using complement

\(P(X \geq 8) = 1 - P(X \leq 7)\)

Using GDC: 1 - binomcdf(10, 0.5, 7)

\(= 1 - 0.9453 = 0.0547\)

\(P(X \geq 8) = 0.0547\) or 5.47%

(d) Expected value and standard deviation:

\(E(X) = np = 10 \times 0.5 = 5\)

\(\sigma = \sqrt{np(1-p)} = \sqrt{10 \times 0.5 \times 0.5} = \sqrt{2.5} = 1.58\)

Expected heads: 5, Standard deviation: 1.58

3. Normal Distribution

The Normal Distribution

Normal Distribution \(X \sim N(\mu, \sigma^2)\)

Notation: \(X \sim N(\mu, \sigma^2)\)

\(\mu\): mean (center of distribution)
\(\sigma^2\): variance
\(\sigma\): standard deviation (spread)

Key Properties:

Bell-shaped, symmetric curve centered at \(\mu\)
Mean = Median = Mode = \(\mu\)
Total area under curve = 1
Approximately 68% of data within 1 standard deviation of mean
Approximately 95% within 2 standard deviations
Approximately 99.7% within 3 standard deviations

Standardization and Z-scores

Standard Normal Distribution

Standard Normal: \(Z \sim N(0, 1)\)

Mean = 0, Standard deviation = 1

Z-score Formula:

\(z = \frac{x - \mu}{\sigma}\)

Converts any normal value to standard normal

Interpretation: Z-score tells how many standard deviations \(x\) is from the mean

Finding Normal Probabilities

Using GDC (Recommended):

normalcdf(lower, upper, \(\mu\), \(\sigma\)): Finds \(P(a \leq X \leq b)\)
For \(P(X < a)\): use normalcdf(-∞, a, \(\mu\), \(\sigma\))
For \(P(X > a)\): use normalcdf(a, ∞, \(\mu\), \(\sigma\))
invNorm(\(p\), \(\mu\), \(\sigma\)): Finds value \(x\) where \(P(X \leq x) = p\)

Note: On calculator, use large negative/positive numbers for -∞ and ∞ (e.g., -9999, 9999)

⚠ Normal Distribution Pitfalls:

Variance vs standard deviation: Notation uses \(\sigma^2\), but GDC uses \(\sigma\)
Parameter order: Check if your GDC wants mean first or last
Sketch first: Draw normal curve to visualize what area you need
Units matter: Z-score is unitless, but original \(x\) has units

Example 2: Normal Distribution (IB-Style)

Problem: Heights of adult males are normally distributed with mean 175 cm and standard deviation 8 cm. Let \(X\) represent height.

(a) Write the distribution of \(X\)

(b) Find the probability a randomly selected male is taller than 180 cm

(d) Find the height that 90% of males are shorter than

Solution:

(a) Distribution:

Mean \(\mu = 175\) cm, Standard deviation \(\sigma = 8\) cm

\(X \sim N(175, 8^2)\) or \(X \sim N(175, 64)\)

(b) \(P(X > 180)\):

Using GDC: normalcdf(180, 9999, 175, 8)

\(P(X > 180) = 0.266\) or 26.6%

Interpretation: About 26.6% of adult males are taller than 180 cm

(c) \(P(170 \leq X \leq 185)\):

Using GDC: normalcdf(170, 185, 175, 8)

\(P(170 \leq X \leq 185) = 0.630\) or 63.0%

(d) Height where 90% are shorter (90th percentile):

We want to find \(x\) where \(P(X \leq x) = 0.90\)

Using GDC: invNorm(0.90, 175, 8)

\(x = 185.25\) cm

Height = 185 cm (or 185.3 cm to 1 d.p.)

Interpretation: 90% of males are shorter than 185.3 cm; only 10% are taller

📋 Probability Distributions Quick Reference

Distribution	Notation	Mean	Variance
General Discrete	\(X\)	\(E(X) = \sum x \cdot P(X=x)\)	\(E(X^2) - [E(X)]^2\)
Binomial	\(X \sim B(n, p)\)	\(np\)	\(np(1-p)\)
Normal	\(X \sim N(\mu, \sigma^2)\)	\(\mu\)	\(\sigma^2\)

🎯 IB Exam Strategy

Common Question Types:

"Find E(X)": Use \(\sum x \cdot P(X=x)\) or formulas for specific distributions
"X follows binomial...": Check BINS, use GDC binompdf/binomcdf
"Normally distributed...": Use normalcdf for probabilities, invNorm for values
"At least/at most": Use complement or cumulative probabilities
"Find k where P(X ≤ k) = ...": Use invNorm for normal, trial and error for binomial

Key Reminders:

Always use GDC for binomial and normal calculations
Check all probabilities sum to 1 for discrete distributions
Binomial: check BINS conditions first
Normal: sketch curve to visualize required area
Expected value formulas: \(np\) for binomial, \(\mu\) for normal
Variance: \(np(1-p)\) for binomial, \(\sigma^2\) for normal

🎉 Master Probability Distributions!

Probability distributions provide powerful mathematical models for random phenomena. The binomial distribution handles fixed-trial experiments, while the normal distribution models countless natural phenomena from heights to test scores. Master these distributions, their formulas, and GDC techniques to excel in IB exams and prepare for advanced statistics and data science!

Key Success Factors:

✓ Discrete: \(E(X) = \sum x \cdot P(X=x)\), Var = \(E(X^2) - [E(X)]^2\)
✓ Binomial: Check BINS conditions, \(X \sim B(n,p)\)
✓ Binomial formulas: \(E(X) = np\), \(\text{Var}(X) = np(1-p)\)
✓ Normal: \(X \sim N(\mu, \sigma^2)\), bell curve, symmetric
✓ Z-score: \(z = \frac{x-\mu}{\sigma}\) standardizes any normal value
✓ Use GDC: binompdf/cdf for binomial, normalcdf/invNorm for normal

Check Conditions • Use GDC • Interpret Results

Master probability distributions and excel in IB Mathematics! 🚀