Geometric Distributions

Front

What is a geometric distribution?

Back

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

Front

What is the probability of success in a geometric distribution?

Back

The probability of success (p) is the likelihood of achieving the desired outcome in a single trial.

Tags

CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

Front

What is the formula for the probability of having the first success on the k-th trial?

Back

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

Front

What does 'at least k trials' mean in the context of geometric distributions?

Back

It refers to the scenario where the first success occurs on or after the k-th trial.

Tags

CCSS.HSF.BF.A.2

Front

How do you calculate the probability of needing at least 3 trials to get the first success?

Back

P(X >= 3) = 1 - P(X < 3) = 1 - (P(X = 1) + P(X = 2)).

Front

What is the expected value of a geometric distribution?

Back

The expected value (mean) is E(X) = 1/p, where p is the probability of success.

Front

What is the variance of a geometric distribution?

Back

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

Front

In the context of the field-goal kicker, what does a 90% success rate imply?

Back

It implies that the probability of making a field goal is 0.9, and the probability of missing is 0.1.

Front

What is the significance of the 'first tails' in a coin flip scenario?

Back

It refers to the trial number at which the first tails appears, which can be modeled using a geometric distribution.

Tags

CCSS.HSF.BF.A.2

Front

How do you interpret the result of 0.9375 in the coin flip question?

Back

It means there is a 93.75% chance that the first tails will occur within the first 4 flips.

Tags

CCSS.HSF.BF.A.2