AP Stats binomial and geometric distributions key terms

What is a binomial distribution?

A binomial distribution models the number of successes in a fixed number of independent trials, each with two possible outcomes (success or failure).

True or False: A binomial experiment requires trials to be dependent.

False. Trials must be independent for a binomial distribution.

Example of a binomial scenario?

Flipping a coin 10 times and counting the number of heads.

Fill in the blank: The number of trials, n, is _____ in a binomial distribution.

fixed, known before the experiment.

What are the two outcomes in a binomial experiment?

Success and failure.

How is the binomial probability formula structured?

$\displaystyle P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$, where $\displaystyle n$ is trials, $\displaystyle k$ is successes.

Compare success probability in binomial vs geometric.

Binomial has a fixed probability, while geometric's success probability remains constant per trial.

What does 'n' represent?

'n' represents the total number of trials in a binomial distribution.

What does 'p' signify?

'p' is the probability of success on each trial.

If n=5, p=0.3, what is the probability of getting 2 successes?

Use $P(X = 2) = \binom{5}{2} (0.3)^2 (0.7)^3 = 0.163.

True or False: The mean of a binomial distribution is np.

True. The mean is calculated as $\displaystyle E(X) = np$.

What is the variance of a binomial distribution?

Variance is calculated as $\displaystyle Var(X) = np(1-p)$.

Example of calculating binomial probabilities?

In a store, the probability of selling a shirt is 0.4. Over 5 days, find probability of selling 3 shirts.

Effect of increasing n on binomial distribution shape?

Increasing 'n' typically makes the distribution more symmetric and bell-shaped.

True or False: Binomial distributions can model more than two outcomes.

False. Binomial distributions only model two outcomes.

What is the range of values for X in a binomial distribution?

X can take on integer values from 0 to n, where n is the total number of trials. - Example: If n=10, X can be 0, 1, 2,..., 10. - Represents the number of successes.

What is a geometric distribution?

A probability distribution modeling the number of trials until the first success occurs.

True or False: Geometric distributions are used for fixed number of trials.

False - They are used for variable trials until the first success.

Example of a geometric situation?

Flipping a coin until you get heads. Each flip is a trial.

Definition of 'success' in context?

The desired outcome, e.g., getting heads in coin flips.

What does 'p' represent?

The probability of success on each trial, e.g., 0.5 for a fair coin.

Calculate the expected number of trials for p=0.2.

Use $\displaystyle E(X) = \\frac{1}{p}$. Therefore, $\displaystyle E(X) = \\frac{1}{0.2} = 5$.

Fill in the blank: The variance of a geometric distribution is ___ .

$\displaystyle Var(X) = \\frac{1-p}{p^2}$.

What is the memoryless property?

The probability of success on future trials does not depend on past trials.

Example of waiting time?

A store customer waiting for the first sale item. Trials are each purchase.

If p=0.1, what is the probability of success on the first trial?

$\displaystyle P(X=1) = p = 0.1$.

True or False: Geometric distributions can only model discrete data.

True - They are suited for discrete trials.

Difference between geometric and binomial distributions?

Geometric counts trials until the first success; binomial counts successes in fixed trials.

What does 'q' represent?

The probability of failure, calculated as $\displaystyle q = 1 - p$.

Example problem: Probability of first success on 4th trial?

$\displaystyle P(X=4) = q^{3}p = (0.9)^{3}(0.1) = 0.729$.

What is cumulative probability in geometric distributions?

The probability of achieving success by a certain number of trials.

Fill in the blank: The probability of no success in n trials is ___.

$\displaystyle P(X > n) = q^n$.

What is the primary focus of a binomial distribution?

Successes in a fixed number of trials.

What is the primary focus of a geometric distribution?

Number of trials until the first success.

Fill in the blank: Binomial distributions require a ____ number of trials.

fixed

True or False: Binomial distributions can have variable trials.

False - Binomial has a fixed number.

List two key characteristics of binomial distributions.

- Fixed number of trials - Two outcomes (success/failure)

List two key characteristics of geometric distributions.

- Variable number of trials - Two outcomes (success/failure)

When is the binomial distribution used?

When you know the number of trials beforehand.

When is the geometric distribution used?

When you're interested in the first success occurrence.

True or False: The mean of a binomial distribution is $\displaystyle n imes p$.

True - where n is trials and p is success probability.

True or False: The mean of a geometric distribution is $\displaystyle \\frac{1}{p}$.

True - where p is the probability of success.

Example: A store sells light bulbs. After 10 purchases, what's the chance of 3 defective bulbs?

Use binomial distribution: n=10, k=3, find P(X=3).

Example: You keep flipping a coin until it lands heads. What distribution is this?

Geometric distribution.

What does 'n' represent in a binomial distribution?

Number of trials.

What does 'p' represent in both distributions?

Probability of success.

Comparison: Binomial vs Geometric distributions in terms of trials.

Binomial has a fixed number of trials; Geometric is variable.

Cause: The first success occurs in a trial series. Effect: Which distribution applies?

Geometric distribution applies.

Scenario: Coin toss 5 times. Find success probability.

Use binomial distribution. P(X=k) = $\displaystyle {n ext{ choose } k} imes p^k imes (1-p)^{n-k}$. Here, n=5, k=number of heads.

True or False: Binomial distribution requires constant probability.

True. In a binomial distribution, the probability of success must remain the same for each trial.

Fill in the blank: Geometric distribution models the number of trials until the first ____.

success. It counts trials until the first success occurs.

Calculate: Probability of 3 successes in 10 trials, p=0.4.

Use binomial formula: P(X=3) = $\displaystyle {10 ext{ choose } 3} imes (0.4)^3 imes (0.6)^7$. Value is approximately 0.215.

Scenario: Rolling a die until a 6 appears.

This is geometric. The expected number of rolls is E(X) = $\displaystyle {\frac{1}{p}}$ where p = \frac{1}{6}.

Comparison: Binomial vs. Geometric distribution.

- Binomial: Fixed number of trials. - Geometric: Trials continue until the first success.

Scenario: Store sells 70% success rate for sales.

If making 8 calls, use P(X=k) to find probability of k sales out of 8.

What does the mean of a binomial distribution represent?

It represents the expected number of successes, calculated by $\displaystyle {E(X) = n imes p}$.

How does the geometric distribution apply to waiting times?

It models the expected number of trials until the first event, such as waiting for a bus.

Scenario: 5% chance of rain each day. 10 days.

Use binomial: Find P(X=k) for k days of rain in 10 days.

True or False: Geometric distribution can have zero trials.

False. At least one trial is needed to achieve the first success.

Calculate expected value for 3 trials, p=0.2.

E(X) for geometric is $\displaystyle {\frac{1}{p}} =${\frac{1}{0.2}} = 5$.

Scenario: Defective items in a batch.

If 2% are defective, find probability of 3 defectives in 100 items using binomial distribution.

What is the variance formula for a binomial distribution?

Variance is calculated as $\displaystyle {Var(X) = n imes p imes (1-p)}$.

Fill in the blank: Mean of geometric distribution is ____.

E(X) = $\displaystyle {\frac{1}{p}}$.

A store sells light bulbs. How many trials until a defect?

This is a geometric distribution scenario. You count the number of trials until the first success (defective bulb). If the probability of a bulb being defective is 0.1, and you want to find the probability of the first defective bulb on the 5th trial, use: $\displaystyle P(X = k) = (1 - p)^{k-1} p$