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 states (success or failure) across a fixed number of trials.

Front

What are the parameters of a binomial distribution?

Back

The parameters of a binomial distribution are 'n' (the number of trials) and 'p' (the probability of success on each trial).

Front

How do you calculate the probability of getting exactly k successes in n trials?

Back

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

Front

What does 'n choose k' mean in the context of binomial distributions?

Back

'n choose k' (denoted as C(n, k) or nCk) is the number of ways to choose k successes from n trials, calculated as n! / (k!(n-k)!).

Front

What is the mean of a binomial distribution?

Back

The mean (expected value) of a binomial distribution is calculated as μ = n * p.

Front

What is the variance of a binomial distribution?

Back

The variance of a binomial distribution is calculated as σ² = n * p * (1 - p).

Front

What is the standard deviation of a binomial distribution?

Back

The standard deviation of a binomial distribution is the square root of the variance: σ = √(n * p * (1 - p)).

Front

In a binomial distribution, what does it mean to have 'at least' k successes?

Back

'At least' k successes means calculating the probability of getting k or more successes, which can be found by summing the probabilities from k to n.

Front

How do you find the probability of getting at least k successes?

Back

To find the probability of getting at least k successes, calculate P(X >= k) = 1 - P(X < k) = 1 - Σ P(X = i) for i = 0 to k-1.

Front

What is the probability of getting exactly 10 heads when flipping a coin 20 times?

Back

The probability is 17.62%.