Discrete Probability Distribution

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Questions and Answers

Which of the following describes a discrete probability distribution?

A distribution that can take on any value within a range.

A model that only accounts for continuous random variables.

A function that lists all possible values of a random variable with their probabilities. (correct)

A function that predicts future events based on past occurrences.

What are the two conditions that a discrete probability distribution must satisfy?

P(X=x) = x/n and P(X=x) = n/x

P(X=x) ≥ 0 and Σ P(X=x) < 1

P(X=x) = x and n > 0

0 ≤ P(X=x) ≤ 1 and Σ P(X=x) = 1 (correct)

In a binomial distribution, what does the variable 'p' represent?

The total number of trials conducted.

The probability of a successful event. (correct)

The probability of failure in the trials.

The average number of successes.

What is the formula for calculating the variance in a binomial distribution?

σ² = np(1 - p) Signup and view all the answers

How do you calculate the probability of getting exactly 4 successes in a binomial experiment with 12 trials?

Using the formula P(X) = nCk * p^k * (1 - p)^(n - k) Signup and view all the answers

What is the mean for a binomial distribution given n trials and probability p of success?

µ = np Signup and view all the answers

What type of distribution describes successes in trials from a finite population without replacement?

Hypergeometric Distribution Signup and view all the answers

If a student randomly guesses on a multiple-choice test with 10 questions, what is the probability of answering exactly 6 questions correctly?

Use the binomial formula considering p = 0.25. Signup and view all the answers

What is the mean number of trials required to achieve r successes in a Negative Binomial Distribution?

$\frac{r}{p}$ Signup and view all the answers

In the context of Negative Binomial Distribution, what does the variance formula represent?

$\frac{r(1 - p)}{p^2}$ Signup and view all the answers

For a survey with a 9% completion rate, what is the probability of getting the 3rd completed survey on the 10th call?

$\binom{9}{2} (0.09)^3 (0.91)^7$ Signup and view all the answers

What is the probability that the first defective tire will be identified on the 27th inspection, if the defect probability is 2%?

$0.02(0.98)^{26}$ Signup and view all the answers

How is the probability of receiving exactly 8 orders in one day calculated if the average is 12 orders?

$\frac{12^8 e^{-12}}{8!}$ Signup and view all the answers

If a startup receives an average of 7 text messages over 3 hours, what is the probability of receiving exactly 9 messages in that same period?

$\frac{7^9 e^{-7}}{9!}$ Signup and view all the answers

What is the probability that the third oil strike occurs on the seventh drilled well, given a 20% success rate?

$\binom{6}{2}(0.2)^3(0.8)^4$ Signup and view all the answers

What is the variance of the number of trials needed to achieve three successful oil strikes when the probability is 20%?

$\frac{3(0.8)}{0.2^2}$ Signup and view all the answers

What does the variable $K$ represent in the probability formula?

Objects classified as successes Signup and view all the answers

In the provided probability formulas, what does the value $N$ signify?

Total objects in the population Signup and view all the answers

If a sample of 10 marbles is drawn from an urn containing 400 red and 600 blue marbles, what is the probability of drawing exactly 3 red marbles using the appropriate formula?

It is calculated using the hypergeometric distribution. Signup and view all the answers

In a geometric distribution, what does the mean ($ ext{µ}$) equal?

$rac{1}{p}$ where $p$ is the probability of success Signup and view all the answers

What is the variance ($ ext{σ}^2$) for a geometric distribution?

$rac{q}{p^2}$ Signup and view all the answers

When considering the probability that the 5th car is the first red car on a Tuesday, which type of probability distribution is being utilized?

Geometric distribution Signup and view all the answers

If the probability of a tire being defective is 2%, what is the probability that the 5th tire tested is a defect?

0.02 Signup and view all the answers

In the given scenario of engineers, what does the variable $x$ signify in the context of the geometric distribution?

The number of trials until the first success Signup and view all the answers

What cumulative geometric distribution formula is used to find the probability that $X$ is less than or equal to $x$?

$1 - q^x$ Signup and view all the answers

To calculate the mean number of tires expected to test until finding the first defective one, which value is used?

The probability of defect $p$ Signup and view all the answers

Study Notes

Probability Distributions

A probability distribution defines possible values of a random variable and their associated probabilities.
Discrete probability distributions cover values of discrete random variables and their likelihoods.

Conditions for Discrete Probability Distribution

Probabilities must satisfy (0 ≤ P(X = x) ≤ 1).
The sum of probabilities must equal (1): (\sum P(X = x) = 1).

Types of Discrete Probability Distributions

Binomial Distribution: Deals with two possible outcomes (success or failure).
- Probability formula: (P(X) = \binom{n}{x} p^x q^{n - x})
- Mean: (\mu = np)
- Variance: (\sigma^2 = np(1 - p))
Hypergeometric Distribution: Describes number of successes in n trials from a finite population without replacement.
- Probability formula: (P(x) = \frac{\binom{K}{x} \binom{N-K}{n-x}}{\binom{N}{n}})
Geometric Distribution: Defines chances of achieving first success in independent and identical trials.
- Probability: (P(X = x) = q^{x - 1} p)
- Cumulative distribution: (P(X ≤ x) = 1 - q^x)
- Mean: (\mu = \frac{1}{p})
- Variance: (\sigma^2 = \frac{q}{p^2})
Negative Binomial Distribution: Counts trials until the r-th success.
- Probability formula: (P(X) = \binom{n - 1}{r - 1} p^r q^{n - r})
Poisson Distribution: Models the number of events occurring in a fixed interval.
- Probability formula: (P(X = x) = \frac{\mu^x e^{-\mu}}{x!})

Applications of Discrete Probability Distributions

Binomial Example: Rolling a die 12 times, calculate the probability of rolling a "4" five times, utilizing binomial formulas.
Hypergeometric Example: Calculate probabilities for students enrolled in a course among a sample size.
Geometric Example: Finding probabilities related to the sequence of car colors.

Specific Probability Problems

Doctor and nurse selection scenario: Calculate probability of choosing 4 doctors from a total of 25 names without replacement.
Marbles example: Find the likelihood of drawing a specific number of red marbles from an urn.
Tire defect analysis: Determine probabilities of defective vs. non-defective tires.

Velocity of Events in Independence

Returning probabilities concerning company operations like identifying defects or strikes in geological studies.

Summary Formulas

Binomial:
- Mean: (\mu = np)
- Variance: (\sigma^2 = np(1-p))
Negative Binomial:
- Mean: (\mu = \frac{r}{p})
- Variance: (\sigma^2 = \frac{rq}{p^2})
Geometric:
- Mean: (\mu = \frac{1}{p})
- Variance: (\sigma^2 = \frac{q}{p^2})
Poisson:
- For ( \mu) events per interval, use formulas for generating probability for x occurrences.

Real-World Applications

Utilization in various industries like risk assessment, quality control, or service efficiency.

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Description

This quiz explores the concept of discrete probability distributions, focusing on the function that describes possible values of a discrete random variable and their associated probabilities. Test your knowledge on the key characteristics and applications of this fundamental concept in probability theory.