# Application of the Binomial Distribution to an Aircraft

Suppose you are flying a four-engine aircraft on a 10-hour transoceanic flight. Assume that it has been established that the probability of an engine of the type you have on your aircraft failing during a 10-hour flight is 0.008. Determine the probability of all four engines failing during the flight.

A student decided that this is a binomial distribution problem. Remember that the four characteristics of a binomial distribution are:

Fixed number of trials

Two possible outcomes

Fixed probability of success

Independence of events

Evaluate the problem in terms of the four characteristics and explain whether you think it does or does not fit the characteristics of a binomial distribution. If not, which characteristics do not hold? Note that in determining whether or not a problem is â??binomialâ? or not, you must evaluate it against the four characteristics listed above.

The student found that the probability of all four engines failing is 4.1E-09. Is that answer correct?

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#### Solution Preview

In general, for n independent trials with a probability p of success for each trial, the probability of m successes is P = nCm * p^m * q^(n-m), where q = 1-p is the probability of failure of a given trial and nCm = n!/(m!(n-m)!) is a binomial coefficient. In this case, we ...

#### Solution Summary

We use the binomial distribution to estimate the probability of failure of a four-engine aircraft on a transoceanic flight, given the probability of failure of any one of its engines.

Probability & Statistics

Some Discrete Probability Distributions - (Binomial & Multinomial Distributions/Hypergeometric Distribution/Negative Binomial & Geometric Distributions/Poisson Distribution)

(See the attached file for full description).

1. In a certain city district the need for money to buy drugs is stated as the reason for 75% of all thefts. Find the probability that among the next 5 theft cases reported in this district,

a) exactly 2 resulted from the need for money to buy drugs

b) at most 3 resulted from the need for money to buy drugs

2. According to a survey by the Administrative Management Society, one-half of US companies give employees 4 weeks of vacation after they have been with the company for 15 yrs. Find the probability that among 6 companies surveyed at random, the number that give employees 4 weeks of vacation after 15 yrs of employment is

a) anywhere from 2 to 5

b) fewer than 3

3. According to a study published by a group of University of Massachusetts sociologists, approximately 60% of the Valium users in the state first took Valium for psychological problems. Find the probability that among the next 8 users interviewed from this state,

a) exactly 3 began taking Valium for psychological problems

b) at least 5 began taking Valium for problems that were not psychological

4. Assuming that 6 in 10 automobile accidents are due mainly to a speed violation; find the probability that among 8 automobile accidents 6 will be due mainly to a speed violation

a) by using the formula for the binomial distribution

b) by using the binomial table

5. To avoid detection at customs, a traveler places 6 narcotic tablets in a bottle containing 9 vitamin pills that are similar in appearance. If the customs official selects 3 of the tablets at random for analysis, what is the probability that the traveler will be arrested for illegal possession of narcotics?

6. What is the probability that a waitress will refuse to serve alcoholic beverages to only 2 minors if she randomly checks the IDs of 5 students from among 9 students of which 4 are not of legal age?

7. A scientist inoculates several mice, one at a time, with a disease germ until he finds 2 that have contracted the disease. If the probability of contracting the disease is 1/6, what is the probability that 8 mice are required?

8. According to a study published by a group of University of Massachusetts sociologists, about two-thirds of the 20 million persons in this country who take Valium are women. Assuming this figure to be a valid estimate, find the probability that on a given day the fifth prescription written by a doctor for Valium is

a) the first prescribing Valium for a woman

b) the third prescribing Valium for a woman

9. A certain areas of the eastern US is, on average, hit by 6 hurricanes a year. Find the probability that for a given year that area will be hit by

a) fewer than 4 hurricanes

b) anywhere from 6 to 8 hurricanes

10. The average number of field mice per acre in a 5-acre wheat field is estimated to be 12. Find the probability that fewer than 7 field mice are found

a) on a given acre

b) on 2 of the next 3 acres inspected

11. The probability that a person will die from a certain respiratory infection is 0.002. Find the probability that fewer than 5 of the next 2000 so infected will die.

12. The probability that a student fails the screening test for scoliosis at a local high school is known to be 0.004. Of the next 1875 students who are screened for scoliosis, find the probability that

a) fewer than 5 fail the test

b) 8,9, or 10 fail the test

13. Changes in airport procedures require considerable planning. Arrival rates of aircraft are important factors that must be taken into account. Suppose small aircraft arrive at a certain airport, according to a Poisson process, at the rate of 6 per hour. Thus the Poisson parameter for arrivals for a period of hours is Mean=6t.

a) What is the probability that exactly 4 small aircraft arrive during a 1 hr period?

b) What is the probability that at least 4 arrive during a 1 hr period?

c) If we define a working day as 12 hrs, what is the probability that at least 75 small aircraft arrive during a day?

14. In airport luggage screening it is known that 3% of people screened have questionable objects in their luggage. What is the probability that a string of 15 people pass through successfully before an individual is caught with a questionable object? What is the expected number in a row that pass through before an individual is stopped?

5 Questions with answers. I would like to understand how these answers were derived.

1. The number of goals scored by a college soccer team follows a Poisson distribution with a mean of 5 goals per game. Find the probability that a randomly selected game would have more than 3 goals scored.

2. A loan officer has indicated that 80% of all loan application forms have zero errors. If 6 forms are selected at random, the standard deviation for the number of forms with at least one error is

3. The director of human resources on a university campus selects two employees for a certain job from a group of six employees. This group consists of one female and five males. What is the probability that the female is selected for one of the jobs?

4. 60% of a population of coffee drinkers is reported to prefer a certain flavor of coffee. If a group of these coffee drinkers is interviewed as to their preference of the different flavors of coffee, what is the probability that exactly 5 of them must be interviewed before the first says that he/she prefers this certain flavor of coffee?

5. Suppose that the number of e-mail messages received by your instructor during an 8 hr period follows a Poisson distribution with a mean of 4 messages per hr. Find the probability that less than 3 messages are received by your instructor during the next 1 hr period.

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