Statistics and Distributions

How to calculate binomial probability?

Q: What topic usually follows the binomial distribution in 11th grade?

Classes typically move next to the mean and standard deviation of a binomial distribution (mu = np, sigma = sqrt(np(1 - p))), followed by normal approximations to the binomial distribution when n is large.

Quick answer

You calculate a binomial probability using the formula P(X = k) = nCk * p^k * (1 - p)^(n - k). First, identify your fixed number of trials n, the number of successes k, and the probability of success on a single trial p. Here, nCk represents the combination formula n! / (k! * (n - k)!).

The topic explained

The four conditions of a binomial experiment

Before running the numbers, check whether your scenario qualifies as a binomial experiment. It must satisfy four requirements, often remembered by the acronym BINS.

First, Binary: each trial has exactly two outcomes, labeled success and failure. Second, Independent: the result of one trial does not change the likelihood of another. Third, Number of trials: the experiment repeats for a fixed number of trials, n. Fourth, Same probability: the probability of success, p, stays identical across all trials.

The formula for exactly k successes

The formula to find the probability of exactly k successes in n trials is P(X = k) = (n choose k) * p^k * (1 - p)^(n - k).

The term (n choose k), written as n! / (k!(n - k)!), counts the number of different sequences in which your k successes can occur. The term p^k gives the probability of those k successes occurring. The term (1 - p)^(n - k) gives the probability that the remaining n - k trials end in failure. Multiplying all three parts together yields the overall probability.

A concrete example: free throws

Suppose a basketball player makes free throws with an 80% success rate (p = 0.80). If she shoots 5 free throws (n = 5), what is the probability she makes exactly 3 (k = 3)?

First, find 5 choose 3: 5! / (3! * 2!) = 10 ways. Next, compute 0.80^3 = 0.512. Then, calculate (1 - 0.80)^(5 - 3) = 0.20^2 = 0.04. Finally, multiply the pieces: 10 * 0.512 * 0.04 = 0.2048. She has a 20.48% chance of making exactly 3 out of 5 shots.

Using the complement for at least or at most

When a problem asks for 'at least one success', calculating each possibility separately takes too long. Instead, use the complement rule: P(at least 1) = 1 - P(0 successes).

Watch out for phrasing. 'At most 2' means k can be 0, 1, or 2. 'Fewer than 2' means k can only be 0 or 1. Writing down the list of accepted values for k before solving prevents frequent errors.

Free printable worksheet

Binomial Probability Practice Sheet

For 11th-grade students. Verify the four binomial conditions, determine n, p, and k, and solve each problem step by step.

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Free printable worksheetlessoons.com

Binomial Probability Practice SheetName:

A fair coin is flipped 4 times. What is n, and what is p for landing on heads?
Calculate the binomial coefficient 6 choose 2.
A fair 6-sided die is rolled 5 times. What is the probability of rolling no sixes at all?
A multiple-choice quiz has 4 questions, each with 5 answer choices. If you guess randomly on all 4, what is the probability of getting exactly 2 correct?
A machine produces parts that are defective 10% of the time. If a batch of 6 parts is inspected, what is the probability of finding exactly 1 defective part?
Using the same machine from item 5 (defect rate 10%, sample of 6), calculate the probability of finding at least 1 defective part using the complement rule.

Common questions

When does sampling without replacement count as a binomial experiment?

Sampling without replacement breaks independence because the pool shrinks. However, if the population is at least 10 times larger than the sample size (the 10% condition), statisticians treat the trials as independent and use the binomial formula without significant error.

What is the difference between binomial pdf and binomial cdf?

A binomial probability density function (pdf) gives the probability of getting exactly k successes. A binomial cumulative distribution function (cdf) calculates the sum of probabilities up to k successes, meaning k or fewer.

What common mistake do students make with combinations?

Students often forget to multiply by the combination term (n choose k), computing only p^k * (1 - p)^(n - k). That gives the probability of one specific order of outcomes, rather than all possible arrangements.

Can n choose k ever be smaller than 1?

No. The combination term counts the number of ways to pick k items from n. It is always a whole number greater than or equal to 1, provided 0 <= k <= n.

What topic usually follows the binomial distribution in 11th grade?

Classes typically move next to the mean and standard deviation of a binomial distribution (mu = np, sigma = sqrt(np(1 - p))), followed by normal approximations to the binomial distribution when n is large.

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