Question

**Bonus Group Project 1: Negative Binomial
Distribution**

Negative Binomial experiment is based on sequences of Bernoulli trials with probability of success p. Let x+m be the number of trials to achieve m successes, and then x has a negative binomial distribution. In summary, negative binomial distribution has the following properties

- Each trial can result in just two possible outcomes. One is called a success and the other is called a failure.
- The trials are independent
- The probability of success, denoted by p, is the same on every trial.
- The experiment consists of m successes, and x+m repeated
trials, and the m
^{th}success occurs at the (x+m)^{th}trial.

- Write down the probability distribution P(x=k), consistent with the notation here
- If you are tossing a regular coin repeatedly, what is the
probability that the 3
^{rd}head occurs at the 6^{th}time you toss it? - Anne is selling girl scot cookies in her neighborhood with 20
houses. She has a target to sell 10 boxes. Suppose each house has a
probability 0.6 to buy one box of her cookies. What is the
probability that she sells the last box at the 15
^{th}house? What is the probability that she exhausts all 20 houses?

Answer #1

Given , X+m is the number of trials to achieve mth success (X be the number of failures before mth success) , with probability of success is p

X follow Negative Binomial distribution

The probability distribution of X is given by

Tossing a coin

Let X be the number of failures before the third head ( or X+3 be the number of trials), m=3 , p= 0.5 (probability of head)

To find P(X=3) =?

Probability that 3rd head occurs at 6th toss = 0.15625

Selling cookies

Let X+ 10 be the number of house she goes to sells 10 boxes

X follow negative Binomial distribution with m=10 , p =0.6

Probability that she sells the last box at 15th house is 0.12396

Probability that she exhausts 20 houses = 0.05857

Match the distribution to the description
Group of answer choices
Bernoulli
Binomial
Geometric
Negative Binomial
Poisson
Counting the number of occurrences of an event in a continuous
interval
the sum of n independent bernoulli trials
given a series of independent bernoulli trials, stop when you
get r successes (where r can be any positive integer)
given a series of independent bernoulli trials, stop when you
get the first success ...

a. In a binomial distribution with 9 trials and a success
probability of 0.4, what would be the probability of a success on
every trial? Round to 4 decimal places.
b. In a binomial distribution with 12 trials and a success
probability of 0.6, what would be the probability of a success on
every trial? Round to 4 decimal places.
c. A binomial distribution has a success probability of 0.7, and
10 trials. What is the probability (rounded to 4...

for a binomial experiment with r successes out of n trials, what
value do we use as a point estimate for the probability of success
p on a single trial?
p=

Consider a binomial experiment with 16 trials and probability
0.65 of success on a single trial.
(a) Use the binomial distribution to find the probability of
exactly 10 successes. (Round your answer to three decimal
places.)
(b) Use the normal distribution to approximate the probability
of exactly 10 successes. (Round your answer to three decimal
places.)

A binomial experiment consists of 800 trials. The probability of
success for each trial is 0.4. What is the probability of obtaining
300?-325 ?successes? Approximate the probability using a normal
distribution.? (This binomial experiment easily passes the?
rule-of-thumb test for approximating a binomial distribution using
a normal? distribution, as you can check. When computing the?
probability, adjust the given interval by extending the range by
0.5 on each? side.)

Assume that a procedure yields a binomial distribution with with
n=8 trials and a probability of success of p=0.90. Use a binomial
probability table to find the probability that the number of
successes x is exactly 4.
1. P(4)= ?

Assume that a procedure yields a binomial distribution with a
trial repeated n = 18 times. Find the probability of X > 4
successes given the probability p = 0.27 of success on a single
trial.
P(X>4)=

Assume that a procedure yields a binomial distribution with a
trial repeated n times. Use the binomial probability formula to
find the probability of x successes given the probability p of
success on a single trial. Round to three decimal places.
n = 7, x = 4 , p = 0.5

Assume that a procedure yields a binomial distribution with a
trial repeated n times. Use the binomial probability formula to
find the probability of x successes given the probability p of
success on a single trial. Round to three decimal places. n = 4, x
= 3, p = 1/6

Assume that a procedure yields a binomial distribution with a
trial repeated n times. Use the binomial probability formula to
find the probability of x successes given the probability p of
success on a single trial.
n=5, x=2, p=0.55
P(2) = (Round to three decimal places as needed.)

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