IB Mathematics AI SL Binomial distribution. Mean and variance MAI Study Notes- New Syllabus

◆ BINOMIAL DISTRIBUTION – \( B(n,p) \)

It is the distribution of a discrete random variable \( X \) which takes on the values:

$ 0, 1, 2, 3, 4, \dots, n $

with probability function:

$ p(x) = \binom{n}{x} p^x (1-p)^{n-x} \quad \text{for} \quad x = 0, 1, 2, 3, \dots, n $

where \( n \) and \( p \) are two parameters. We will see that the binomial distribution describes a certain type of problems.

SUCCESS with probability \( p \).
FAILURE with probability \( 1-p \).

 \( n \) = number of trials.
 \( p \) = probability of success.

While:

 \( X \) counts the number of (possible) successes.

\( X \sim B(n,p) \).

Since \( n \) is the number of trials, \( X \) can take on the values:

$ 0, 1, 2, 3, 4, \dots, n $

The probabilities \( P(X=0), P(X=1), P(X=2) \), etc., can be obtained by the GDC.

$ \mathbf{GDC}$

Our GDC (Casio) gives the results for a Binomial distribution:

$\text{MENU → Statistics → DIST → BINOMIAL:}$ We use Bpd or Bcd.

For simplicity, let us denote by:

Bpd(x): The probability of exactly \( x \) successes.
Bcd(x₁ to x₂): The probability from \( x₁ \) up to \( x₂ \) successes.

The menu for both functions is:

Data: Always Variable.
Numtrial: The number of trials, i.e., \( n \).
p: The probability of success \( p \) (for each game).

Then, for each value of \( x \) (or \( x₁ \) to \( x₂ \)), EXE gives the result.