Binomial Distribution Reference Cheat Sheet

A printable reference covering binomial conditions, probability formulas, mean, variance, standard deviation, cumulative probability, and normal approximation for grades 10-12.

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The binomial distribution models the number of successes in a fixed number of repeated trials. Students need this reference because binomial problems appear often in probability, statistics, biology, business, and test preparation. This cheat sheet helps you identify when a situation is binomial and choose the correct formula quickly. The most important ideas are the four binomial conditions, the probability formula, and the meaning of the parameters $n$ and $p$ . A binomial random variable is written $X \sim B(n,p)$ , where $n$ is the number of trials and $p$ is the probability of success on each trial. Key summary formulas include the mean $\mu = np$ , variance $\sigma^2 = np(1-p)$ , and standard deviation $\sigma = \sqrt{np(1-p)}$ .

Key Facts

A binomial setting has a fixed number of trials $n$ , only two outcomes per trial, a constant success probability $p$ , and independent trials.
If $X$ counts the number of successes in $n$ trials with success probability $p$ , then $X \sim B(n,p)$ .
The binomial probability formula is $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$ for $k=0,1,2,\ldots,n$ .
The combination formula is $\binom{n}{k}=\frac{n!}{k!(n-k)!}$ , which counts the number of ways to choose $k$ successes from $n$ trials.
The mean of a binomial distribution is $\mu = np$ .
The variance of a binomial distribution is $\sigma^2 = np(1-p)$ .
The standard deviation of a binomial distribution is $\sigma = \sqrt{np(1-p)}$ .
A normal approximation is often reasonable when $np \ge 10$ and $n(1-p) \ge 10$ .

Vocabulary

Binomial distribution: A probability distribution for the number of successes in a fixed number of independent trials with the same success probability.
Trial: One repetition of a random experiment, such as flipping a coin once or asking one person a survey question.
Success: The outcome being counted in a binomial problem, even if it is not a positive or desirable result.
Parameter: A number that defines a probability model, such as $n$ for the number of trials and $p$ for the probability of success.
Combination: A count of selections where order does not matter, written as $\binom{n}{k}$ .
Cumulative probability: The probability of getting a value within a range, such as $P(X \le k)$ or $P(X \ge k)$ .

Common Mistakes to Avoid

Using the binomial formula when trials are not independent is wrong because the probability of success must stay the same from trial to trial.
Confusing $p$ and $1-p$ changes the meaning of success and failure, so always define the success outcome before substituting into $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$ .
Forgetting the combination factor $\binom{n}{k}$ gives the probability of one specific order only, not all possible orders with $k$ successes.
Using $P(X=k)$ when the question asks for at least or at most is wrong because cumulative wording requires adding several probabilities, such as $P(X \le k)$ or $P(X \ge k)$ .
Applying the normal approximation when $np < 10$ or $n(1-p) < 10$ can be inaccurate because the binomial distribution may be too skewed.

Practice Questions

1 A fair coin is flipped $8$ times. If $X$ is the number of heads, find $P(X=5)$ .
2 A basketball player makes $70\%$ of free throws. If the player takes $10$ free throws, find the mean and standard deviation of the number made.
3 A multiple-choice quiz has $12$ questions with $4$ choices each. If a student guesses on every question, what is $P(X \ge 10)$ , where $X$ is the number correct?
4 A factory samples items without replacement from a small batch. Explain why this situation may not satisfy the binomial conditions.

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