A distribution is a way to describe how probabilities are spread across different outcomes. Chapter 6 focuses on the Binomial Distribution.

The Four Conditions (BINS)

You can only use a Binomial model if the situation meets these four criteria:

• B - Binary: There are only two possible outcomes (e.g., Success or Failure, Heads or Tails).
• I - Independent: The result of one trial does not affect the next.
• N - Fixed Number: There is a set number of trials ($n$).
• S - Same Probability: The probability of success ($p$) stays the same for every trial.

Notation

We write a Binomial distribution as:

$X \sim B(n, p)$

• $X$: The random variable (what we are counting).
• $n$: The number of trials.
• $p$: The probability of success.

Calculating Probabilities

Individual Probabilities

To find the chance of exactly $r$ successes, use the formula:

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

or just use Distributions > Binary CD/PD.

Cumulative Probabilities

This is the “running total” of probabilities.

• $P(X \leq x)$: The probability of $x$ or fewer successes. You usually find this using a calculator or a probability table.
• $P(X > x)$: Calculated as $1 - P(X \leq x)$.
• $P(X \geq x)$: Calculated as $1 - P(X \leq x-1)$.

Mean and Variance

For a Binomial distribution, the average (mean) number of successes you expect is simply the number of trials multiplied by the probability:

$\text{Mean } (E[X]) = np$

Discrete Uniform Distribution

The chapter also briefly mentions this. It is when every possible outcome has the exact same probability (like rolling a fair 6-sided die, where every number has a $1/6$ chance).

Binomial CD/PD

• Binomial PD (Probability Distribution): Use this when you want the probability of an exact number (e.g., $X = 6$).
• Binomial CD (Cumulative Distribution): Use this for a range of values (e.g., “at most 6,” “less than 6,” or “more than 6”).