Binomial Distributions

Front

What is a binomial distribution?

Back

A binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent outcomes under a given set of parameters.

Front

What are the key characteristics of a binomial experiment?

Back

1. The number of trials (n) is fixed. 2. Each trial has only two possible outcomes (success or failure). 3. The trials are independent. 4. The probability of success (p) is the same for each trial.

Front

What is the formula for calculating the probability of exactly k successes in a binomial distribution?

Back

P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where (n choose k) = n! / (k!(n-k)!)

Front

How do you calculate the mean of a binomial distribution?

Back

Mean (μ) = n * p, where n is the number of trials and p is the probability of success.

Front

How do you calculate the variance of a binomial distribution?

Back

Variance (σ²) = n * p * (1 - p), where n is the number of trials and p is the probability of success.

Front

What is the probability of getting at least k successes in a binomial distribution?

Back

P(X ≥ k) = 1 - P(X < k) = 1 - Σ P(X = i) for i = 0 to k-1.

Front

What does it mean for trials to be independent in a binomial experiment?

Back

It means the outcome of one trial does not affect the outcome of another trial.

Front

Can a binomial distribution have a probability of success greater than 1?

Back

No, the probability of success (p) must be between 0 and 1, inclusive.

Front

What is the probability of getting exactly 3 successes in 6 trials with a success probability of 0.5?

Back

P(X = 3) = (6 choose 3) * (0.5)^3 * (0.5)^(6-3) = 0.3125.

Front

What is the cumulative distribution function (CDF) for a binomial distribution?

Back

The CDF gives the probability that a binomial random variable is less than or equal to a certain value k.