The Binomial Distribution models the number of successes in a fixed number of independent Bernoulli trials. If you flip a coin $n$ times, and each flip has a probability $p$ of coming up heads, the total number of heads follows a Binomial distribution.

Parameters

• $n$: The fixed number of independent trials.
• $p$: The probability of success on any given single trial.
• $X$: The random variable representing the total number of successes across all $n$ trials.
• $k$: The specific number of successes we want to find the probability for (where $0 \le k \le n$).

Core Properties

• Mean ($E[X]$): $np$
• Variance ($Var(X)$): $np(1-p)$
• Support: $k \in \{0, 1, 2, \dots, n\}$

Advanced Practice

Example 1: The Biased Coin

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Suppose you have a biased coin that lands on heads 70% of the time. If you flip this coin 8 times, what is the probability that you get exactly 6 heads?

Show Solution

Here, the number of trials $n = 8$, the probability of success $p = 0.7$, and the desired number of successes $k = 6$.

We apply the Binomial formula:
$P(X = 6) = \binom{8}{6} (0.7)^6 (1-0.7)^{8-6}$

First, calculate $\binom{8}{6} = \frac{8!}{6!2!} = 28$.
Then, $(0.7)^6 \approx 0.117649$ and $(0.3)^2 = 0.09$.

Multiplying these together: $P(X = 6) = 28 \times 0.117649 \times 0.09 \approx 0.296$.

So there is roughly a 29.6% chance of getting exactly 6 heads.

Example 2: Quality Assurance Limits

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A factory produces lightbulbs with a 5% defect rate. In a random sample of 10 bulbs, what is the probability that at most 1 bulb is defective?

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Let $X$ be the number of defective bulbs. We want to find $P(X \le 1)$, which is $P(X = 0) + P(X = 1)$. Here $n=10$ and $p=0.05$.

First, find $P(X = 0)$:
$P(X = 0) = \binom{10}{0} (0.05)^0 (0.95)^{10} = 1 \times 1 \times (0.95)^{10} \approx 0.5987$

Next, find $P(X = 1)$:
$P(X = 1) = \binom{10}{1} (0.05)^1 (0.95)^9 = 10 \times 0.05 \times (0.95)^9 \approx 0.3151$

Now, sum the probabilities:
$P(X \le 1) \approx 0.5987 + 0.3151 = 0.9138$

There is about a 91.4% chance that at most 1 bulb is defective.

The Range Rule of Thumb

In many distributions, including the Binomial distribution (especially when $np \ge 5$ and $n(1-p) \ge 5$), the vast majority of outcomes will fall within 2 standard deviations of the mean. We use the Range Rule of Thumb to identify "unusual" values.

• Maximum usual value: $\mu + 2\sigma$
• Minimum usual value: $\mu - 2\sigma$

If a value falls outside this range, it is considered statistically unusual.

Example 3: Spotting the Unusual

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A standard fair coin is flipped 100 times, and it lands on heads 65 times. Is getting 65 heads considered an unusual result for a fair coin?

Show Solution

For a fair coin, $p = 0.5$. The number of trials is $n = 100$.

First, calculate the mean ($\mu$) and standard deviation ($\sigma$):
$\mu = np = 100 \times 0.5 = 50$
$\sigma = \sqrt{np(1-p)} = \sqrt{100 \times 0.5 \times 0.5} = \sqrt{25} = 5$

Now, apply the Range Rule of Thumb:
- Maximum usual value: $\mu + 2\sigma = 50 + 2(5) = 60$
- Minimum usual value: $\mu - 2\sigma = 50 - 2(5) = 40$

Usual results are between 40 and 60 heads. Since 65 is strictly greater than the maximum usual value (60), yes, getting 65 heads is considered an unusual result.

The Binomial distribution sits at the center of a family of discrete distributions. Changing its core assumptions (independence, two outcomes, fixed $n$) leads to other well-known distributions.

Discrete Distribution Relationships

How the Binomial distribution relates to other key models.

1. From Bernoulli to Binomial

A Binomial random variable is simply the sum of $n$ independent and identically distributed (i.i.d.) Bernoulli random variables. If $n=1$, the Binomial distribution reduces exactly to the Bernoulli distribution.

2. Binomial vs. Hypergeometric

The Binomial distribution assumes trials are independent (sampling with replacement). If you sample without replacement from a finite population, the probability of success changes with each draw. This scenario is modeled by the Hypergeometric Distribution.

3. Binomial vs. Multinomial

The Binomial distribution deals with exactly two outcomes (Success or Failure). If each trial can result in one of $k$ distinct categories (like rolling a 6-sided die), the distribution of the counts of each category follows a Multinomial Distribution.

4. Binomial vs. Geometric

While Binomial counts the number of successes in a fixed number of trials, the Geometric Distribution counts the number of trials needed to get exactly one success. Both rely on independent Bernoulli trials, but they flip what is fixed and what is random.

5. Binomial vs. Poisson

If $n$ gets extremely large and $p$ gets extremely small, while the mean $np = \lambda$ remains constant, the Binomial distribution converges to the Poisson Distribution. This is useful for modeling rare events.