The binomial distribution is used when a situation has a fixed number of repeated trials and each trial has only two possible outcomes, often called success and failure. It helps answer questions like how likely it is to get exactly 3 heads in 5 coin flips or exactly 8 correct guesses on a quiz. The setup matters because it separates counting the possible arrangements from calculating the probability of each arrangement.
This makes many probability problems organized and predictable.
Key Facts
- Binomial probability formula: P(X = k) = C(n, k)p^k(1 - p)^(n - k)
- Combination formula: C(n, k) = n! / (k!(n - k)!)
- n is the fixed number of trials, and k is the number of successes being counted.
- p is the probability of success on one trial, and 1 - p is the probability of failure.
- The binomial model requires independent trials with the same probability p on every trial.
- The coefficients C(n, k) appear in row n of Pascal's triangle.
Vocabulary
- Binomial distribution
- A probability distribution that gives the chance of getting exactly k successes in n independent two-outcome trials.
- Trial
- One repeated action or experiment, such as flipping a coin once or answering one multiple-choice question.
- Success
- The outcome being counted in a binomial problem, whether or not it is a positive result in everyday language.
- Combination
- A count of how many ways k items can be chosen from n items when order does not matter.
- Pascal's triangle
- A triangular number pattern whose rows give the binomial coefficients used in expansion and binomial probability.
Common Mistakes to Avoid
- Using the binomial formula when the trials are not independent. If one outcome changes the probability of the next outcome, the standard binomial setup does not apply.
- Forgetting the combination factor C(n, k). The factor counts the different orders in which the k successes can occur, so leaving it out usually gives only one arrangement.
- Confusing p with 1 - p. The exponent k must go with the probability of success p, while n - k must go with the probability of failure.
- Treating k as any value instead of a whole number from 0 to n. The number of successes must be an integer count, so values like 2.5 successes are not valid.
Practice Questions
- 1 A fair coin is flipped 6 times. Use the binomial formula to find the probability of getting exactly 4 heads.
- 2 A basketball player makes 70% of free throws. If the player shoots 5 free throws, what is the probability of making exactly 3?
- 3 A bag contains 4 red marbles and 6 blue marbles, and one marble is drawn 5 times without replacement. Explain whether a binomial distribution is appropriate and why.