1. A negative binomial experiment consists of repeated trials that result in one of two outcomes (success/failure). It continues until a fixed number (k) of successes occur.
2. The probability of success (p) is constant across trials, which are independent. The number of trials (x) needed to achieve k successes follows a negative binomial distribution.
3. The document provides the notation and formula for the negative binomial distribution. It also gives examples of calculating the probability of achieving k successes in x trials under this distribution.
Negative Binomial Distribution introduction & over view under the complementary Statistics syllabus of University of Calicut in BSc core of Mathematics, Physics & Computer Science.
Negative Binomial Distribution introduction & over view under the complementary Statistics syllabus of University of Calicut in BSc core of Mathematics, Physics & Computer Science.
A Probability Distribution is a way to shape the sample data to make predictions and draw conclusions about an entire population. It refers to the frequency at which some events or experiments occur. It helps finding all the possible values a random variable can take between the minimum and maximum statistically possible values.
A Probability Distribution is a way to shape the sample data to make predictions and draw conclusions about an entire population. It refers to the frequency at which some events or experiments occur. It helps finding all the possible values a random variable can take between the minimum and maximum statistically possible values.
JEE Mathematics/ Lakshmikanta Satapathy/ Theory of probability part 10/ Bernoulli trials and Binomial distribution of probability of Bernoulli trials and probability function with example
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Chapter 5: Discrete Probability Distribution
5.2 - Binomial Probability Distributions
Topic Covered in this video:
1. What is Discrete Probability Distribution
2. Types of Theoretical Discrete Probability Distribution
3. Binomial Distribution
4. Properties of Binomial Distribution
5. Examples of Binomial distribution
6. Fitting of Binomial Distribution
7. Application of Binomial distribution
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2. Negative Binomial Experiment
A negative binomial experiment is a statistical
experiment that has the following properties:
1-The experiment consists of x repeated trials.
2-Each trial can result in just two possible outcomes. We
call one of these outcomes a success and the other, a
failure.
3-The probability of success, denoted by P, is the same on
every trial.
4-The trials are independent; that is, the outcome on one
trial does not affect the outcome on other trials.
5-The experiment continues until k successes are observed,
where k is specified in advance.
3. Consider the following statistical experiment. You flip a coin
repeatedly and count the number of times the coin lands on
heads. You continue flipping the coin until it has landed 3
times on heads. This is a negative binomial experiment
because:
1-The experiment consists of repeated trials. We flip a coin
repeatedly until it has landed 3 times on heads.
2-Each trial can result in just two possible outcomes –
heads or tails.
3-The probability of success is constant ( 0.5) on every trial.
4-The trials are independent; that is, getting heads on one
trial does not affect whether we get heads on other trials.
5-The experiment continues until a fixed number of
successes have occurred; in this case, 3 heads.
4. Notation
The following notation is helpful, when we talk about
negative binomial probability.
x: The number of trials required to produce k successes in
a negative binomial experiment.
k: The number of successes in the negative binomial
experiment.
P: The probability of success on an individual trial.
q: The probability of failure on an individual trial.
5. Negative Binomial Random Variable
The number X of trials to produce k successes in a negative binomial
experiment is called a negative binomial random variable.
Negative Binomial Distribution
A negative binomial random variable is the number X of repeated trials
to produce k successes in a negative binomial experiment.
The probability distribution of a negative binomial random variable is
called a negative binomial distribution. The negative binomial
distribution is also known as the Pascal distribution.
𝑃 𝑋 = 𝑥 =
𝑥 − 1
𝑘 − 1
𝑝 𝑘
𝑞 𝑥−𝑘 , 𝑓𝑜𝑟 𝑥 = 𝑘, 𝑘+1, 𝑘+2,……
6. Example-1
You are surveying people exiting from a polling booth and asking
them if they voted independent. The probability that a person voted
independent is 25%. What is the probability that 15 people must be
asked before you can find 5 people who voted independent?
Solution
K = 5, x = 15, p = 0.25, q = 0.75
𝑃 𝑋 = 𝑥 =
𝑥 − 1
𝑘 − 1
𝑝 𝑘
𝑞 𝑥−𝑘
𝑃 𝑋 = 15 =
15 − 1
5 − 1
(0.25)5 (0.75)15−5
𝑃 𝑋 = 15 = 0.0552
7. Example-2
Suppose that the probability is 0.8 that any given person will believe
a tale about life after death. What is the probability that the sixth
person to hear this tale is the fourth one to believe it ?
Solution
K = 4, x = 6, p = 0.8, q = 0.2
𝑃 𝑋 = 𝑥 =
𝑥 − 1
𝑘 − 1
𝑝 𝑘
𝑞 𝑥−𝑘
𝑃 𝑋 = 6 =
6 − 1
4 − 1
(0.8)6 (0.2)6−4
𝑃 𝑋 = 6 = 0.1049
8. Example-3
A football player, his success rate of goal hitting is 70%. What is the
probability that player hits his third goal on his fifth attempt?
Solution
K = 3, x = 5, p = 0.7, q = 0.3
𝑃 𝑋 = 𝑥 =
𝑥 − 1
𝑘 − 1
𝑝 𝑘 𝑞 𝑥−𝑘
𝑃 𝑋 = 5 =
5 − 1
3 − 1
(0.7)3 (0.3)5−3
𝑃 𝑋 = 5 = 0.1852
9. Example-4
You draw cards from a deck (with replacement) until you get four
kings. What is the probability that you will draw exactly 20 times
Solution
K = 4, x = 20, 𝑝 =
4
52
= 0.077, q = 0.923
𝑃 𝑋 = 𝑥 =
𝑥 − 1
𝑘 − 1
𝑝 𝑘 𝑞 𝑥−𝑘
𝑃 𝑋 = 20 =
20 − 1
4 − 1
(0.077)4
(0.923)20−4
𝑃 𝑋 = 20 = 0.00945
10. Example-5
Find the probability that a person tossing 3coins will get either all
heads or all tails for the second time on the fifth toss
Solution
K = 2, x = 5, 𝑝 =
2
8
= 0.25, q = 0.75
𝑃 𝑋 = 𝑥 =
𝑥 − 1
𝑘 − 1
𝑝 𝑘
𝑞 𝑥−𝑘
𝑃 𝑋 = 5 =
5 − 1
2 − 1
(0.25)2 (0.75)5−2
𝑃 𝑋 = 5 = 0.1445