Probability Distributions (Binomial and Normal)

Discrete vs Continuous Random Variables

A discrete random variable takes specific, countable values (e.g., number of heads in 10 coin flips). A continuous random variable can take any value in an interval (e.g., height, temperature).

• The binomial distribution is discrete.
• The normal distribution is continuous.

The Binomial Distribution

A random variable $X$ follows a binomial distribution, written $X \sim B(n, p)$, if:

1. There are a fixed number $n$ of independent trials
2. Each trial has exactly two outcomes: success (probability $p$) or failure (probability $q = 1 - p$)
3. The probability of success $p$ is constant for each trial
4. The trials are independent

The probability of exactly $k$ successes is:

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

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

This formula is in your IB formula booklet, but in practice you will almost always use your GDC.

Key statistics:

• Expected value (mean): $E(X) = np$
• Variance: $\text{Var}(X) = np(1-p)$
• Standard deviation: $\sigma = \sqrt{np(1-p)}$

The expected value $E(X) = np$ is particularly important and frequently tested.

GDC Techniques for Binomial

Your GDC has two key functions for binomial calculations:

• Binomial PDF (probability density function): $P(X = k)$ — the probability of exactly $k$ successes
• Binomial CDF (cumulative distribution function): $P(X \leq k)$ — the probability of at most $k$ successes

Using these two functions, you can calculate any binomial probability:

Probability	GDC approach
$P(X = k)$	binomPDF($n, p, k$)
$P(X \leq k)$	binomCDF($n, p, k$)
$P(X < k)$	binomCDF($n, p, k-1$)
$P(X \geq k)$	$1 -$ binomCDF($n, p, k-1$)
$P(X > k)$	$1 -$ binomCDF($n, p, k$)
$P(a \leq X \leq b)$	binomCDF($n, p, b$) $-$ binomCDF($n, p, a-1$)

The Normal Distribution

A continuous random variable $X$ follows a normal distribution, written $X \sim N(\mu, \sigma^2)$, where $\mu$ is the mean and $\sigma$ is the standard deviation.

Properties of the normal distribution:

• The curve is bell-shaped and symmetric about $\mu$
• The mean, median, and mode are all equal to $\mu$
• Approximately $68\%$ of data falls within $\mu \pm \sigma$
• Approximately $95\%$ falls within $\mu \pm 2\sigma$
• Approximately $99.7\%$ falls within $\mu \pm 3\sigma$
• The total area under the curve equals 1

Standardisation and Z-scores

The standard normal distribution has $\mu = 0$ and $\sigma = 1$, written $Z \sim N(0, 1)$.

Any normal variable can be standardised using the z-score formula:

$z = \frac{x - \mu}{\sigma}$

The z-score tells you how many standard deviations a value is from the mean. A z-score of 2 means the value is 2 standard deviations above the mean.

While you can use z-scores and standard normal tables, your GDC can work with any normal distribution directly — you do not need to standardise first on Paper 2.

GDC Techniques for Normal

Your GDC provides:

• Normal CDF: $P(a < X < b)$ for $X \sim N(\mu, \sigma^2)$

  • For $P(X < b)$: use lower bound $= -10^{99}$ (or equivalent)
  • For $P(X > a)$: use upper bound $= 10^{99}$

• Inverse Normal: Given a probability $p$, find the value $x$ such that $P(X < x) = p$

  • This is used for questions like "find the value below which 90% of the data falls"

Inverse Normal Problems

Inverse normal problems give you a probability and ask you to find the corresponding $x$-value. The key is ensuring your probability is a left-tail probability (i.e., $P(X < x)$) before using the inverse normal function.

Example: If $X \sim N(50, 8^2)$ and $P(X > k) = 0.2$, find $k$.

Since $P(X > k) = 0.2$, we have $P(X < k) = 0.8$.

Using inverse normal with area $= 0.8$, $\mu = 50$, $\sigma = 8$: $k = 56.7$ (3 s.f.)

Combining Both Distributions

Some IB questions require you to recognise which distribution to use. Ask yourself:

• Am I counting successes in fixed trials? → Binomial
• Am I measuring a continuous quantity? → Normal
• Does the question mention "probability of success," "independent trials"? → Binomial
• Does it mention "mean and standard deviation" of a measured quantity? → Normal

Tip 1: Check the Conditions for Binomial

Before using the binomial distribution, verify all four conditions (fixed $n$, two outcomes, constant $p$, independence). If any condition is violated, the binomial model may not be appropriate.

Tip 2: Sketch a Normal Curve

For normal distribution questions, always sketch a bell curve, mark the mean, and shade the required area. This helps you identify whether you need a left-tail, right-tail, or central probability, and prevents errors when using the GDC.

Tip 3: Convert Probability Statements Carefully

The most common source of error is misinterpreting inequality signs. $P(X \geq 5)$ is NOT the same as $P(X > 5)$ for discrete distributions. For continuous distributions, $P(X \geq 5) = P(X > 5)$ since the probability at a single point is zero.

Tip 4: Use Complement for "At Least" Problems

For binomial problems: $P(X \geq k) = 1 - P(X \leq k-1) = 1 - \text{binomCDF}(n, p, k-1)$. The complement approach is usually easier than summing individual probabilities.

Tip 5: Write Down Your GDC Input

In the exam, write exactly what you enter into your GDC (e.g., "binomCDF(10, 0.3, 4) = 0.850"). This shows your method clearly and earns marks even if you make a keying error.