Geometric Distribution

Front

What is a geometric distribution?

Back

A geometric distribution models the number of trials needed to achieve the first success in a series of independent Bernoulli trials, where each trial has the same probability of success.

Front

What is the formula for the probability of the first success on the k-th trial in a geometric distribution?

Back

P(X = k) = (1 - p)^(k-1) * p, where p is the probability of success.

Front

If the probability of success is 0.25, what is the probability of failing the first two trials and succeeding on the third?

Back

P(X = 3) = (1 - 0.25)^(2) * 0.25 = 0.14.

Front

What does it mean if a process follows a geometric distribution?

Back

It means that the process consists of repeated independent trials until the first success occurs.

Front

How do you calculate the probability of more than k trials before the first success?

Back

P(X > k) = (1 - p)^k, where p is the probability of success.

Front

In a geometric distribution, what is the expected number of trials until the first success?

Back

The expected number of trials is E(X) = 1/p.

Front

What is the variance of a geometric distribution?

Back

The variance is Var(X) = (1 - p) / p^2.

Front

How does the geometric distribution relate to Bernoulli trials?

Back

The geometric distribution is derived from a series of independent Bernoulli trials, where each trial has two possible outcomes: success or failure.

Front

What is the probability of having no successes in n trials if the success probability is p?

Back

P(no successes in n trials) = (1 - p)^n.

Front

If a player has a 67% success rate, what is the probability that they will fail their first four attempts?

Back

P(fail first 4) = (1 - 0.67)^4 = 0.01.