When Is the Binomial Distribution an Appropriate Model?

A binomial distribution is appropriate when there is a fixed number of independent trials, each trial has exactly two possible outcomes, and the probability of success remains constant. If these conditions hold, write $X \sim B(n,p)$, where $n$ is the number of trials and $p$ is the probability of success.

This is SL 4.8: Binomial distribution in IB Mathematics: Applications and Interpretation.

The Four Conditions

For example, suppose a basketball player makes each free throw independently with probability $0.7$, and takes $10$ shots. If $X$ is the number of successful shots, then

$$X \sim B(10,0.7).$$

The model is appropriate because there are $10$ fixed trials, each shot is either made or missed, the shots are independent, and the success probability remains $0.7$.

It would not be appropriate if the player became tired so that the success probability changed after each shot. Likewise, drawing cards without replacement usually violates independence and the constant-probability condition.

A common misconception is that any experiment with two outcomes is binomial. Two outcomes alone are insufficient; all four conditions must hold.

Exam Technique

In an IB response asking whether a binomial model is appropriate, apply the command term justify by referring explicitly to the context: fixed $n$, two outcomes, independence, and constant $p$. For probability calculations, state the distribution before using the GDC, and give the final answer exactly or to three significant figures unless instructed otherwise. A GDC is permitted on every Math AI paper.