12.1 Formula of the Binomial Distribution

The binomial distribution is a fundamental probability distribution that describes the probability of obtaining a specific number of successes in a fixed number of independent Bernoulli trials. A Bernoulli trial is an experiment with only two possible outcomes: success or failure.

Formula

The probability of getting exactly $x$ successes in $n$ independent Bernoulli trials, where the probability of success in a single trial is $p$, is given by the binomial probability formula:

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

Alternatively, using $q$ to represent the probability of failure:

$$ P(X = x) = \binom{n}{x} p^x q^{n-x} $$

Where:

• $P(X = x)$: The probability of observing exactly $x$ successes.
• $\binom{n}{x}$: The binomial coefficient, often read as "n choose x". It represents the number of ways to choose $x$ successes from $n$ trials without regard to the order. It is calculated as: $$ \binom{n}{x} = \frac{n!}{x!(n-x)!} $$
  • $n!$ (n factorial) is the product of all positive integers up to $n$ ($n! = n \times (n-1) \times \dots \times 2 \times 1$).
  • $x!$ is the factorial of $x$.
  • $(n-x)!$ is the factorial of $(n-x)$.
• $p$: The probability of success in a single independent trial.
• $q$: The probability of failure in a single independent trial. It is always true that $q = 1 - p$.
• $n$: The total number of independent trials.
• $x$: The number of successes we are interested in. The possible values for $x$ range from 0 to $n$ ($x \in {0, 1, 2, \dots, n}$).

Key Concepts

• Independent Trials: Each trial's outcome does not affect the outcome of any other trial.
• Two Possible Outcomes: Each trial results in either success or failure.
• Constant Probability of Success: The probability of success ($p$) remains the same for every trial.

Example: Coin Toss

Consider flipping a fair coin 5 times ($n=5$). The probability of getting heads (success) in a single flip is $p=0.5$. Consequently, the probability of getting tails (failure) is $q = 1 - 0.5 = 0.5$.

Let's calculate the probability of getting exactly 2 heads ($x=2$) in these 5 flips.

Using the formula:

$$ P(X = 2) = \binom{5}{2} (0.5)^2 (0.5)^{5-2} $$

First, calculate the binomial coefficient $\binom{5}{2}$:

$$ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(3 \times 2 \times 1)} = \frac{120}{(2)(6)} = \frac{120}{12} = 10 $$

Now, substitute this back into the probability formula:

$$ P(X = 2) = 10 \times (0.5)^2 \times (0.5)^3 $$ $$ P(X = 2) = 10 \times 0.25 \times 0.125 $$ $$ P(X = 2) = 10 \times 0.03125 $$ $$ P(X = 2) = 0.3125 $$

Therefore, the probability of getting exactly 2 heads in 5 coin flips is 0.3125 or 31.25%.

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Frequently Asked Questions

What is the formula for the binomial distribution probability?

The formula is $P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$, where $n$ is the number of trials, $x$ is the number of successes, and $p$ is the probability of success in a single trial.

How do you interpret each term in the binomial distribution formula?

• $\binom{n}{x}$: The number of different sequences of $n$ trials that contain exactly $x$ successes.
• $p^x$: The probability of achieving $x$ successes.
• $(1-p)^{n-x}$: The probability of achieving $(n-x)$ failures.

What does nCx represent in the binomial formula?

$nCx$ (or $\binom{n}{x}$) represents the binomial coefficient, which quantifies the number of distinct ways to choose $x$ successes from a set of $n$ trials.

How do you calculate the binomial coefficient?

The binomial coefficient $\binom{n}{x}$ is calculated using the formula $\frac{n!}{x!(n-x)!}$.

What is the difference between p and q in binomial distribution?

$p$ is the probability of success in a single trial, while $q$ is the probability of failure in a single trial. They are related by $q = 1 - p$.

Can you explain how the binomial distribution applies to a coin toss experiment?

A coin toss experiment fits the binomial distribution criteria if:

1. You perform a fixed number of tosses (n).
2. Each toss has only two outcomes: heads (success) or tails (failure).
3. The probability of heads (p) is constant for each toss.
4. The tosses are independent of each other.

How do you calculate the probability of exactly x successes in n trials?

You use the binomial probability formula: $P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$.

Why are Bernoulli trials important in binomial distribution?

The binomial distribution is fundamentally built upon a sequence of independent Bernoulli trials, each with the same probability of success. This structure is essential for the formula to be applicable.

What values can x take in a binomial distribution?

The number of successes, $x$, can take any integer value from 0 up to the total number of trials, $n$. So, $x \in {0, 1, 2, \dots, n}$.

How would you solve for P(X = 2) when n = 5 and p = 0.5?

You would use the binomial probability formula: $P(X = 2) = \binom{5}{2} (0.5)^2 (1-0.5)^{5-2}$ $P(X = 2) = \frac{5!}{2!3!} \times (0.5)^2 \times (0.5)^3$ $P(X = 2) = 10 \times 0.25 \times 0.125 = 0.3125$