information about the occurrence of the antecedent, p. Then we can draw an inductive conclusion about the probability of the occurrence of the consequent, q.
In the example below, we use the conditional from the deductive reasoning section and add a probabilistic premise:
If there is a credible bomb threat to a Federal office building, then the building is always evacuated until a search is completed.
A Federal office building has a 1% chance of receiving a credible bomb threat sometime within a year.
Therefore, a Federal office building has at least a 1% chance of being evacuated during a year’s time.
The conclusion has the probability value of at least 1% because the building may have some likelihood of being evacuated for reasons other than a bomb threat. That probability is not contained in the information that our premises give us, but we must allow for it in our conclusion.
Now let us vary this formula by combining the conditional with information about the probability that the consequent is true. Suppose we have information that “the chance that a Federal office building will be evacuated for a search at some point during the year is 1%.” We can then conclude that the chance that a credible bomb threat is made to a Federal building is no greater than 1%. This formula can be expressed as follows:
If there is a credible bomb threat to a Federal office building, then the building is always evacuated until a search is completed.
The chance that a Federal office building will be evacuated for a search at some point during the year is 1%.
Therefore, the chance that a credible bomb threat will be made to a Federal building is no greater than 1% in a year.
This conclusion is true because the 1% chance of evacuation includes evacuations due to credible bomb threats and any other causes. The conditional statement tells us that there will not be a credible bomb threat without an evacuation of the building. So, the upper limit on the chance of a bomb threat is 1%.
When a conditional is combined with an added probabilistic premise about either the antecedent or the consequent, there is always a conclusion that has a range of probabilities. We will illustrate the remaining conclusions through the use of formulas based on the example above. (Further real-life examples of these formulas will be found in the Self-
Test.) The first is the formula that applies when you know the probability that the antecedent is not true:
If there is a credible bomb threat to a Federal office building, then the building is always evacuated until a search is completed.
A Federal office building has a 99% chance of not receiving a credible bomb threat sometime within a year.
Therefore, a Federal office building has no more than a 99% chance of not being evacuated during a year’s time.
Since the building has a 99% chance of not receiving a credible bomb threat, it has a 1% chance of receiving such a threat and thus being evacuated. This relationship puts an upper limit of 99% on the probability that the building will not be evacuated. In addition, there is some (unknown) chance of its being evacuated for some other reason. Therefore, the probability that the building will not be evacuated might be lower than 99%.
The next formula applies when you know the probability that the consequent is not true: If there is a credible bomb threat to a Federal office building, then the building is always evacuated until a search is completed.
The chance that a Federal office building will not be evacuated for a search at some point during the year is 99%.
Therefore, the chance that a credible bomb threat will not be made to a Federal building is equal to or greater than 99% in a year.
If the second premise is true, there is a 1% chance that the building will be evacuated for a search at some point during the year. This means that there is no more than a 1% chance that a credible bomb threat will be made and thus no less than a 99% chance that such a bomb threat will not be made.
Biconditionals
When a biconditional is used with an added premise containing a probability, the resulting conclusion has a specific probability rather than a range of probability. Let us illustrate this with the following biconditional statement:
The Secretary of DHS is the director of your agency if and only if you are an employee of DHS.
There is a 60% chance that a person attending this conference is an employee of DHS.
Therefore, there is a 60% chance that the Secretary of DHS is the agency director of a person attending this conference.
This formula is reversible; in other words, you can begin with information about the probability that the Secretary of DHS is the agency director of a person attending this conference and conclude that the person has the same probability of being an employee of DHS.
You can also draw conclusions if you have information that the antecedent or the consequent is not true. For example, suppose you have the information that “There is a 30% chance that a student in this computer training course is not an employee of DHS.” From that information, you could conclude that “There is a 30% chance that the Secretary of DHS is not the agency director of a student in this computer training course.” This type of formula will be used in exercise 3 in the Self-Test.
Advanced Topic: Other Connectives
Extended connectives based on the conditional can be used with added probabilistic premises. So can compound connectives that are based on the conditional (see exercise 4 in the Self-Test). However, the other connectives, such as alternations and conjunctions, cannot be adapted so easily because they entail computations with the probabilities. This topic will be discussed in the unit on statistical reasoning.
Self-Test: Section III.C.2 (answers are given on page 128)
1. The following paragraph contains a conditional premise and an added premise with a probability. Given the information in the paragraph, decide which of conclusions a through c are correct.
Paragraph
In one district, the Detention and Deportation staff had a busy caseload of criminal and noncriminal alien cases in which deportation procedures were pending. If an alien in one of these cases was a criminal, he or she was not detained at the DHS detention facility. Instead, criminal aliens were placed in county jails. In this district, one-tenth of all cases concerned criminal aliens.
Conclusions
a. A case in this district had at least a 10% chance of not involving someone who was detained at the DHS detention facility. (Correct / incorrect)
b. A case in this district had no more than a 90% chance of involving someone who was detained at the DHS detention facility. (Correct / incorrect)
c. A case in this district had less than a 90% chance of concerning a noncriminal alien. (Correct / incorrect)
2. The following paragraph contains a conditional premise and an added premise with a probability. Given the information in the paragraph, decide which of conclusions a through c are correct.
Paragraph
If an employee contributes to the Combined Federal Campaign through payroll deduction, there is a record of a special deduction on the employee’s biweekly earnings statement. In a certain agency, there was a 90% chance that any randomly selected employee had a record of some kind of special deduction on his or her earnings statement.
Conclusions
a. There is at least a 10% chance that a randomly selected employee does not contribute to the Combined Federal Campaign through payroll deduction.
(Correct / incorrect)
b. There is a 10% or greater chance that a randomly selected employee did contribute to the Combined Federal Campaign through payroll deduction.
(Correct / incorrect)
c. There is a 10% chance that a randomly selected employee did not have a special deduction on his or her biweekly earnings statement. (Correct / incorrect)
3. The following paragraph contains a biconditional premise and an added premise with a probability. Given the information in the paragraph, decide which of conclusions a through c are correct.
Paragraph
Employees in a certain category are eligible for overtime pay if and only if they work more than eight hours in one day. Records showed that one employee in this category worked more than eight hours on one-fourth of her regular work days in the last year.
Conclusions
a. There was more than a 25% chance that this employee was eligible for overtime pay on any randomly selected day. (Correct / incorrect)
b. There was a 75% chance that this employee was not eligible for overtime pay on any randomly selected day. (Correct / incorrect)
c. The chance that this employee was eligible for overtime pay on any randomly selected day was 25%. (Correct / incorrect)
4. Advanced Exercise: The following paragraph contains a compound conditional premise and an added premise with a probability. Given the information in the paragraph, decide which of conclusions a through c are correct.
Paragraph
In a study of 500 work stations that had problems with LAN access, it was decided that if the work stations got certain new hardware or upgraded software, the problems would be corrected. After six months had elapsed, it was found that for 40% of the work stations, the problems had not been corrected.
Conclusions
a. For any given work station, there was a 20% chance that it had not gotten new hardware and a 20% chance that it had not gotten upgraded software.
(Correct / incorrect)
b. For any given work station, the probability was 60% that it had gotten both the new hardware and the upgraded software. (Correct / incorrect)
c. For any given work station, the probability was at least .4 that it had gotten neither the specified new hardware nor the upgraded software. (Correct / incorrect)