Math

StatisticsQ&A LibraryJoe is playing an arcade game and has 8 chances to knock down the moving ducks. A person playing this game hits the moving duck 78% of the time.What is the probability that Joe hits the 4th duck?What is the probability that the first duck Joe hits is not until his fourth attempt?What is the probability that Joe will hit 5 ducks?Six ducks at most?At least 4 ducks?What is the probability that he will hit between 4 and 7 ducks inclusive?What is the probability that he will hit at leat one duck?Question

Asked Mar 22, 2019

24 views

Joe is playing an arcade game and has 8 chances to knock down the moving ducks. A person playing this game hits the moving duck 78% of the time.

What is the probability that Joe hits the 4th duck?

What is the probability that the first duck Joe hits is not until his fourth attempt?

What is the probability that Joe will hit 5 ducks?

Six ducks at most?

At least 4 ducks?

What is the probability that he will hit between 4 and 7 ducks inclusive?

What is the probability that he will hit at leat one duck?

Step 1

**1.**

**Find the probability that Joe hits the fourth duck :**

A person playing the Arcade game hits the moving duck 78% of the time.

That is, the probability of hitting the moving duck is *p *= 0.78.

The probability of not hitting the moving duck is *q *= 1 – *p *= 0.22.

The number of chances to knock down the moving ducks is *n *= 8.

Here, hitting the ducks in 8 chances or hits will not be dependent on the other, imp-lying independence among each other. As a result, the 8 chances or hits may be considered as independent trials.

Here, all the 8 hits have a probability of 0.78 of hitting the duck. That is, the rate of success (hitting the duck) is constant. Hence, the rate of failure (not hitting the duck) is also constant.

Therefore, the probability of hitting any duck will be 0.78.

Thus, the probability that Joe hits the fourth duck is **0.78.**

Step 2

**Introduction to geometric distribution:**

Consider a random experiment involving repeated *n *independent Bernoulli trials until a success is obtained.

Moreover, the probability of getting a success in each *n *independent trial, *p*, remains a constant for all the trials. Denote the probability of failure as *q*. As success and failure are mutually exclusive, *q *= 1 – *p*.

Let the random variable *X* denote the number of trails before the first success. Thus, *X* can take any of the values 0,1,2,….*n.*

Then, the probability distribution of *X* is a Geometric distribution with parameter (*p*) and the probability mass function (pmf) of *X*, that is, of a geometric random variable, is given as:

Step 3

**2.**

**Find the probability that the first duck Joe hits is not until the fourth attempt :**

Consider the event of hitting the moving duck as a “success”.

Consider *X* as the number of chances used before hitting the first moving duck, Then, *X* has a geometric distribution with parameter (*p* = 0.78) and the pmf of *X* is

*P*(*x*) = (078)*(0.22)*x*

It is given that the first duck Joe hits...

Tagged in

Find answers to questions asked by student like you

Show more Q&A

Q: The scores on a test are normally distributed with a mean of 150 and a standard deviation of 30. Fin...

A: The normal distribution:A continuous random variable X is said to follow normal distribution if the ...

Q: You are a graduate student in an M.D./Ph.D program, attempting a study of the efficacy of different ...

A: Hello. Since your question has multiple sub-parts, we will solve the first three sub-parts for you. ...

Q: 2. In a lab experiment, students estim is known to have a true value of 5% weight by volume, but stu...

A: Let X be the concentration estimate. X follow a normal distribution with mean 5% and standard deviat...

Q: The body temperatures of a group of healthy adults have a bell-shaped distribution with a mean of 98...

A: According to the empirical rule, in the standard normal curve, approximately 68% of the area is with...

Q: D, E

A: d.The null and alternative hypotheses are:

Q: I created a Venn Diagram to Solve This Problem.

A: Given Total number of tourists = 1232Total number of tourists visited Magic kingdom = 300Total numbe...

Q: Consider a drug that is used to help prevent blood clots in certain patients. In clinical trials, am...

A: Test statistic calculation:The test hypotheses are:Null hypothesis:H0: p = 0.03, that is, 3% in the ...

Q: The land area of th estate of Texas is about 261,797 square miles. If in mid-2016 all the world's pe...

A: The land area of Texas is about 261,797 square miles.Total number of people in world is 7.4 billion....

Q: A person with a cough is a persona non grata on airplanes, elevators, or at the theater. In theaters...

A: (a)Poisson distribution is most suitable, when the events are “rare” and occur independently of each...