Binomial Distribution in Math AA SL
Understanding the Binomial Distribution
The binomial distribution is a probability distribution that models the number of successes—and by extension, failures—in a fixed number of independent trials, each with the same probability of success.
NoteThe binomial distribution is applicable when an experiment meets the following criteria:
- There is a fixed number of trials (n)
- Each trial is independent
- There are only two possible outcomes for each trial (success or failure)
- The probability of success (p) remains constant for all trials
Key Parameters
The binomial distribution is characterized by two parameters:
- n: the number of trials
- p: the probability of success on each trial
These parameters are crucial in determining the shape and properties of the distribution.
Probability Mass Function
The probability of exactly $k$ successes in $n$ trials is given by the binomial probability mass function:
Consider an event that occurs $n$ times, with probability of success $p$ and probability of failure $1-p$. If we want the probaility where we get $k$ number of successes. Let us call the the event of sucesss $S$ and the event of failure $F$.
Because there are multiple different orders that this can happen, for example:
$$SSSFF \cdots$$
$$SFSS \cdots$$
so we just need to determine the number of ways we can order this.
This process of choosing the number of ways to order is exactly the same as the binomial coefficient of $(a+b)^n$.
Therefore, since the binomial coefficient is given by \binom{nCk}, we can say that
$$ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} $$
Where:
- $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$
