2.3 Problems on Minterm Analysis

Consider the class $\left\{A , B , C , D\right\}$ of events. Suppose the probability that at least one of the events A or C occurs is 0.75 and the probability that at least one of the four events occurs is 0.90. Determine the probability that neither of the events A or C but at least one of the events B or D occurs.

Suppose $P \left(A \cup B^{c} C\right) = 0 . 65 , P \left(A C\right) = 0 . 2 , P \left(A^{c} B\right) = 0 . 25$

$P \left(A^{c} C^{c}\right) = 0 . 25 , P \left(B C^{c}\right) = 0 . 30$. Determine $P \left(\left(A C^{c} \cup A^{c} C\right)B^{c}\right)$.

Then determine $P \left(\left(A B^{c} \cup A^{c}\right)C^{c}\right)$ and $P \left(A^{c} \left(B \cup C^{c}\right)\right)$, if possible.

Suppose $P \left(\left(A B^{c} \cup A^{c} B\right)C\right) = 0 . 4 , P \left(A B\right) = 0 . 2 , P \left(A^{c} C^{c}\right) = 0 . 3 , P \left(A\right) = 0 . 6 ,$$P \left(C\right) = 0 . 5 ,$ and $P \left(A B^{c} C^{c}\right) = 0 . 1$. Determine $P \left(A^{c} C^{c} \cup A C\right)$, $P \left(\left(A B^{c} \cup A^{c}\right)C^{c}\right)$, and $P \left(A^{c} \left(B \cup C^{c}\right)\right)$, if possible.

Suppose $P \left(A\right) = 0 . 6 , P \left(C\right) = 0 . 4 , P \left(A C\right) = 0 . 3 , P \left(A^{c} B\right) = 0 . 2 ,$

and $P \left(A^{c} B^{c} C^{c}\right) = 0 . 1$.

Determine $P \left(\left(A \cup B\right)C^{c}\right)$, $P \left(A C^{c} \cup A^{c} C\right)$, and $P \left(A C^{c} \cup A^{c} B\right)$, if possible.

Suppose $P \left(A\right) = 0 . 5 , P \left(A B\right) = P \left(A C\right) = 0 . 3 , \text{and} P \left(A B C^{c}\right) = 0 . 1$.

Determine $P \left(A\left(B C^{c}\right)^{c}\right)$ and $P \left(A B \cup A C \cup B C\right)$.

Then repeat with additional data $P \left(A^{c} B^{c} C^{c}\right) = 0 . 1$ and $P \left(A^{c} B C\right) = 0 . 05$.

Given: $P \left(A\right) = 0 . 6 , P \left(A^{c} B^{c}\right) = 0 . 2 , P \left(A C^{c}\right) = 0 . 4$, and $P \left(A C D^{c}\right) = 0 . 1$.

Determine $P \left(A^{c} B \cup A \left(C^{c} \cup D\right)\right)$.

A survey of a represenative group of students yields the following information:

1. 52 percent are male
2. 85 percent live on campus
3. 78 percent are male or are active in intramural sports (or both)
4. 30 percent live on campus but are not active in sports
5. 32 percent are male, live on campus, and are active in sports
6. 8 percent are male and live off campus
7. 17 percent are male students inactive in sports

1. …

A survey of 100 persons of voting age reveals that 60 are male, 30 of whom do not identify with a political party; 50 are members of a political party; 20 nonmembers of a party voted in the last election, 10 of whom are female. How many nonmembers of a political party did not vote? Suggestion Express the numbers as a fraction, and treat as probabilities.

During a period of unsettled weather, let A be the event of rain in Austin, B be the event of rain in Houston, and C be the event of rain in San Antonio. Suppose:

1. What is the probability of rain in all three cities?
2. What is the probability of rain in exactly two of the three cities?
3. What is the probability of rain in exactly one of the cities?

One hundred students are questioned about their course of study and plans for graduate study. Let $A =$ the event the student is male; $B =$ the event the student is studying engineering; $C =$ the event the student plans at least one year of foreign language; $D =$ the event the student is planning graduate study (including professional school). The results of the survey are:

There are 55 men students; 23 e…

A survey of 100 students shows that: 60 are men students; 55 students live on campus, 25 of whom are women; 40 read the student newspaper regularly, 25 of whom are women; 70 consider themselves reasonably active in student affairs—50 of these live on campus; 35 of the reasonably active students read the newspaper regularly; All women who live on campus and 5 who live off campus consider the…

A television station runs a telephone survey to determine how many persons in its primary viewing area have watched three recent special programs, which we call a, b, and c. Of the 1000 persons surveyed, the results are:

221 have seen at least a; 209 have seen at least b; 112 have seen at least c; 197 have seen at least two of the programs; 45 have seen all three; 62 have seen at least a a…

An automobile safety inspection station found that in 1000 cars tested:

1. 100 needed wheel alignment, brake repair, and headlight adjustment
2. 325 needed at least two of these three items
3. 125 needed headlight and brake work
4. 550 needed at wheel alignment

1. How many needed only wheel alignment?
2. How many who do not need wheel alignment need one or none of the other items?

Suppose $P \left(A\left(B \cup C\right)\right) = 0 . 3 , P \left(A^{c}\right) = 0 . 6 , \text{and} P \left(A^{c} B^{c} C^{c}\right) = 0 . 1$.

Determine $P \left(B \cup C\right)$, $P \left(\left(A B \cup A^{c} B^{c}\right)C^{c}\cupAC\right)$, and $P \left(A^{c}\left(B \cup C^{c}\right)\right)$, if possible.

Repeat the problem with the additional data $P \left(A^{c} B C\right) = 0 . 2$ and $P \left(A^{c} B\right) = 0 . 3$.

A computer store sells computers, monitors, printers. A customer enters the store. Let A, B, C be the respective events the customer buys a computer, a monitor, a printer. Assume the following probabilities:

1. $$ The probability $P \left(A B\right)$ of buying both a computer and a monitor is 0.49.
2. $$ The probability $P \left(A B C^{c}\right)$ of buying both a computer and a monitor but not a printer is 0.17.
3. $$ The probability $P \left(A C\right)$ of buyi…

Data are $P \left(A\right) = 0 . 232$, $P \left(B\right) = 0 . 228$, $P \left(A B C\right) = 0 . 045$, $P \left(A C\right) = 0 . 062$,

$P \left(A B \cup A C \cup B C\right) = 0 . 197$ and $P \left(B C\right) = 2 P \left(A C\right)$.

Determine $P \left(A \cup B \cup C\right)$ and $P \left(A^{c} B^{c} C\right)$, if possible.

Repeat, with the additional data $P \left(C\right) = 0 . 230$.

Data are: $P \left(A\right) = 0 . 4 , P \left(A B\right) = 0 . 3 , P \left(A B C\right) = 0 . 25 , P \left(C\right) = 0 . 65 ,$

$P \left(A^{c} C^{c}\right) = 0 . 3$. Determine available minterm probabilities and the following,

if computable:

With only six items of data (including $P \left(\Omega\right) = P \left(A \vee A^{c}\right) = 1$), not all minterms are available. Try the additional data $P \left(A^{c} B C^{c}\right) = 0 . 1$ and $P \left(A^{c} B^{c}\right) = 0 . 3$. Are these consistent and linearly independent? Are all minterm probabilities available?

Repeat the exercise with $P \left(A B\right)$ changed from 0.3 to 0.5. What is the result? Explain the reason for this result.