The binomial distribution answers one of the most practical probability questions: how likely is a certain number of successes in a fixed number of tries? Coin flips, quiz guesses, and quality checks all follow it. You can check your work with the free tools on Calc Solver.
When the Binomial Distribution Applies
Four conditions must all hold. There is a fixed number of trials, each trial has exactly two outcomes (success or failure), the success probability is the same every trial, and trials are independent, so one result does not affect the next. Flipping a fair coin 10 times meets all four; drawing cards without replacement does not, because probabilities change.
The Formula
The probability of exactly k successes in n trials is: C(n, k) times p^k times (1 – p)^(n – k). Here C(n, k) counts the ways to choose which trials succeed, p is the success probability, and (1 – p) is the failure probability. Each part of the formula has a clear job: counting, successes, and failures.
Worked Example
Find the probability of exactly 3 heads in 5 flips of a fair coin. Here n = 5, k = 3, and p = 0.5. C(5, 3) is 10, p^3 is 0.125, and (1 – p)^2 is 0.25. Multiply: 10 times 0.125 times 0.25 equals 0.3125, so the probability is about 31 percent.
Mean and Spread
The mean number of successes is n times p. For 100 free throws at 80 percent accuracy, expect 80 makes. The standard deviation is the square root of n times p times (1 – p), which measures how much results typically vary around that mean. Mean and spread turn the whole distribution into two useful numbers.
Common Mistakes to Avoid
First, using the binomial formula when trials are not independent or probabilities change. Second, confusing “exactly k” with “at most k” or “at least k”, which need sums of several terms. Third, forgetting that C(n, k) counts orderings: without it, the answer is far too small.
Practice Problems to Try
Find the probability of exactly 2 sixes in 4 rolls of a fair die, then the probability of at least 8 correct guesses in 10 true-or-false questions answered randomly. State the mean and standard deviation for 50 attempts at 60 percent success.