Find probabilities using the binomial distribution

Learning Objectives Identify the four conditions of a binomial experiment. Calculate the binomial coefficient C(n, k) and understand its role as a rational expression of factorials. Apply the binomial probability formula to find the probability of exactly 'k' successes in 'n' trials. Calculate cumulative probabilities, such as P(X ≤ k) or P(X ≥ k). Determine the mean (expected value) and standard deviation of a binomial distribution. Model and solve real-world problems using the binomial distribution. Ever wondered about the probability of passing a 10-question multiple-choice quiz just by guessing? 🤔 The binomial distribution gives us the exact tool to find out! In our study of rational functions, we've focused on expressions defined by ratios. We...

TermDefinitionExample Bernoulli TrialA single random experiment with exactly two possible outcomes: 'success' or 'failure'. The probability of success is denoted by 'p', and the probability of failure is '1-p'.Flipping a single coin. 'Heads' can be a success (p=0.5), and 'Tails' a failure (1-p=0.5). Binomial ExperimentAn experiment that consists of a fixed number of independent Bernoulli trials, where the probability of success remains constant for each trial.Flipping a coin 10 times and counting the number of heads. Here, n=10, and each flip is an independent Bernoulli trial. Binomial Random Variable (X)A variable that counts the number of successes 'k' in a binomial experiment with 'n' trials.If we roll a st...

Binomial Probability Formula P(X=k) = C(n, k) \cdot p^k \cdot (1-p)^{n-k} Use this formula to find the probability of getting exactly 'k' successes in 'n' trials. Here, C(n, k) is the binomial coefficient, 'p' is the probability of success, and 'n' is the number of trials. Binomial Coefficient Formula C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!} This is the formula for 'n choose k'. It calculates the number of combinations and acts as the coefficient in the binomial probability formula. Note its structure as a rational expression. Mean (Expected Value) of a Binomial Distribution \mu = E(X) = np Calculates the average number of successes you would expect in a binomial experiment over the long run. Standard Deviatio...