Binomial Distributions

Front

What is a binomial random variable?

Back

A binomial random variable is a variable that counts the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

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 mean of a binomial distribution?

Back

The mean (expected value) of a binomial distribution is calculated using the formula: μ = n * p.

Front

How do you calculate the variance of a binomial distribution?

Back

The variance of a binomial distribution is calculated using the formula: σ² = n * p * (1 - p).

Front

What is the formula for the probability of getting exactly k successes in n trials?

Back

The formula is 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) represents the number of ways to choose k successes from n trials.

Front

What is the standard deviation of a binomial distribution?

Back

The standard deviation is calculated as σ = √(n * p * (1 - p)).

Front

If the probability of success is 0.32 and there are 75 trials, what is the standard deviation?

Back

The standard deviation is approximately 4.24.

Front

What is the probability of getting at least k successes in n trials?

Back

To find the probability of getting at least k successes, sum the probabilities of getting k, k+1, ..., n successes.

Front

How do you find the probability of getting exactly 3 correct answers when guessing on 6 multiple choice questions with 4 options?

Back

Use the binomial formula with n=6, k=3, and p=0.25 (1/4). The probability is approximately 0.132.