SL 4.8—Binomial distribution Notes

Binomial Distribution

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.

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 $\frac{n!}{k!(n-k)!}$
• $p^k$ represents the probability of k successes
• $(1-p)^{n-k}$ represents the probability of (n-k) failures

Mean and Variance

Two important properties of the binomial distribution are its mean (expected value) and variance:

1. Mean: $\mu = np$
2. Variance: $\sigma^2 = np(1-p)$

Using Technology for Binomial Probabilities

Students are expected to use technology (such as graphing calculators or statistical software) to calculate binomial probabilities. This typically involves inputting the parameters n and p, along with the specific value or range of k for which the probability is needed.

Cumulative Binomial Probability

Often, we're interested in the probability of getting up to a certain number of successes, rather than exactly a specific number. This is where cumulative binomial probability comes in.

Recognizing Binomial Situations

A crucial skill for students is identifying scenarios where the binomial distribution is an appropriate model. Here are some key indicators:

1. Fixed number of trials
2. Two possible outcomes per trial (success/failure)
3. Constant probability of success
4. Independent trials