Solution Found!
Consider the population of voters described in Example
Chapter 3, Problem 35E(choose chapter or problem)
Problem 35E
Consider the population of voters described in Example 3.6. Suppose that there are N = 5000 voters in the population, 40% of whom favor Jones. Identify the event favors Jones as a success S. It is evident that the probability of S on trial 1 is .40. Consider the event B that S occurs on the second trial. Then B can occur two ways: The first two trials are both successes or the first trial is a failure and the second is a success. Show that P( B) = .4. What is P( B| the first trial is S)? Does this conditional probability differ markedly from P( B)?
Reference
Suppose that 40% of a large population of registered voters favor candidate Jones. A random sample of n = 10 voters will be selected, and Y, the number favoring Jones, is to be observed. Does this experiment meet the requirements of a binomial experiment?
If each of the ten people is selected at random from the population, then we have ten nearly identical trials, with each trial resulting in a person either favoring Jones (S) or not favoring Jones (F). The random variable of interest is then the number of successes in the ten trials. For the first person selected, the probability of favoring Jones (S) is .4. But what can be said about the unconditional probability that the second person will favor Jones? In Exercise 3.35 you will show that unconditionally the probability that the second person favors Jones is also .4. Thus, the probability of a success S stays the same from trial to trial. However, the conditional probability of a success on later trials depends on the number of successes in the previous trials. If the population of voters is large, removal of one person will not substantially change the fraction of voters favoring Jones, and the conditional probability that the second person favors Jones will be very close to .4. In general, if the population is large and the sample size is relatively small, the conditional probability of success on a later trial given the number of successes on the previous trials will stay approximately the same regardless of the outcomes on previous trials. Thus, the trials will be approximately independent and so sampling problems of this type are approximately binomial.
Questions & Answers
QUESTION:
Problem 35E
Consider the population of voters described in Example 3.6. Suppose that there are N = 5000 voters in the population, 40% of whom favor Jones. Identify the event favors Jones as a success S. It is evident that the probability of S on trial 1 is .40. Consider the event B that S occurs on the second trial. Then B can occur two ways: The first two trials are both successes or the first trial is a failure and the second is a success. Show that P( B) = .4. What is P( B| the first trial is S)? Does this conditional probability differ markedly from P( B)?
Reference
Suppose that 40% of a large population of registered voters favor candidate Jones. A random sample of n = 10 voters will be selected, and Y, the number favoring Jones, is to be observed. Does this experiment meet the requirements of a binomial experiment?
If each of the ten people is selected at random from the population, then we have ten nearly identical trials, with each trial resulting in a person either favoring Jones (S) or not favoring Jones (F). The random variable of interest is then the number of successes in the ten trials. For the first person selected, the probability of favoring Jones (S) is .4. But what can be said about the unconditional probability that the second person will favor Jones? In Exercise 3.35 you will show that unconditionally the probability that the second person favors Jones is also .4. Thus, the probability of a success S stays the same from trial to trial. However, the conditional probability of a success on later trials depends on the number of successes in the previous trials. If the population of voters is large, removal of one person will not substantially change the fraction of voters favoring Jones, and the conditional probability that the second person favors Jones will be very close to .4. In general, if the population is large and the sample size is relatively small, the conditional probability of success on a later trial given the number of successes on the previous trials will stay approximately the same regardless of the outcomes on previous trials. Thus, the trials will be approximately independent and so sampling problems of this type are approximately binomial.
ANSWER:
Answer:
Step 1 of 3:
(a)
There are voters in the population, 40% of whom favor Jones. The event favors Jones as a success
it is evident that the probability of
on trial one is
. consider the event
that
occurs on the second trial.
The can occur two ways: The first two trials are both successes or the first trial is a failure and the second is a success.
We need to show that
The event can be written like this,
…………(1)
Hence the probability,
…………..(2)
Where and
is the success in the first and second trial respectively and
is the failure in the first trial.
(i). We will calculate the value of
Probability of success in the first trial is = [given in the question]
In the second trial number of voters, will reduce to
, since we are not replacing any voter here.
Hence the probability of success in the second trial is =
Now we can solve,
……(3)
(ii). We will calculate the value of
Probability of failure in the first trial is =
In the second trial number of voters, will reduce to
, since we are not replacing any voter here.
Hence the probability of success in the second trial is,
Now we can solve,
……(4)
Using equation (3) and (4), we can write,
………..(5)
Hence proved