Probability and Statistics: Solved Problems

Posted on Feb 17, 2025 in Mathematics

Assignment A

1. Electrical Unit Failure Analysis

An electrical unit consists of four components, each subject to failure. We observe this unit at a specified time to determine which components are working and which have failed. What is the sample space associated with this random experiment?

Let: denote working, and denote failed.

The sample space is:

2. Set Operations

Suppose the sample space is: . Let: , , and .

Determine the sets:

Here:

Thus,

So,

3. Natural Numbers Divisibility

Let: , the set of natural numbers divisible by 3, and the set of natural numbers divisible by .

What is the set ? What is the set ? and

So, and

4. Newspaper Subscriptions

In a certain residential suburb, of all households subscribe to the metropolitan newspaper, subscribe to the local paper, and subscribe to both newspapers. What proportion of households subscribe to exactly one of the two newspapers?

Out of households, subscribe to both papers. Thus, subscribe only to the metropolitan paper and subscribe only to the local paper. Hence, subscribe to exactly one of the two newspapers.

5. Picture Placement Designs

There are nine different locations in which a picture can be added to a text. If four different pictures are to be placed in the text, how many different designs are there?

Since the pictures are different, this is a permutation problem. So, the number of different designs are:

6. Sample Selection

In how many ways can a sample containing non-defective and defective parts be chosen from a group containing non-defective and defective parts?

There are ways of choosing six non-defective parts from the group of twenty-two non-defective parts and ways of choosing two defective parts from the group of seven defective parts. So, the total number of choices (by the multiplication principle) would be:

7. Barcode Combinations

How many barcodes can be formed using five ‘s, four ‘s and seven ‘s?

Here we use the principle of similar permutations with and . Of course, . So the required number is:

8. Selection with and without Order

Suppose the number of ways of selecting items from a group when order is important is . What would this number be if the order was not important?

Thus, if and , the number would be:

Assignment B

1. Dice Roll Probability

Two fair, distinct dice are rolled. What is the probability that the first die comes up 1 given that the sum on the two dice is 5?

Let be the event the first die is 1, and let be the event the second die is 5. We want:

2-4. Marital Status Data Analysis

Use this information for problems 2, 3, and 4: The following data on the marital status of U.S. adults was found in Current Population Reports:

	Single	Married	Widowed	Divorced	Total
Male	129	298	13	40	480
Female	104	305	57	54	520
Total	233	603	70	94	1000

2. Probability Calculations

;

Find

According to this sample, 60.3% were married, 48% were male, and 5.2% were divorced females.

3. Conditional Probabilities

(a)

(b)

(c) Probability of being widowed given that the person is a male is 14.6%. The probability that the person is a female given that the person is divorced is 10.4%.

4. Independence of Events

Are and independent events? Justify your answer.

; ;

Since , the events are dependent.

5. Dice Roll Independence

Two fair, distinct dice (one red and one green) are rolled. Let be the event the red die comes up even and be the event the sum on the two dice is even. Are independent events?

By listing the elements in each event we see that: ; and . Thus, the events are independent.

6. Congressional Member Probability

For the 107th Congress, 18.7% of the members were senators and 50% of the senators were Democrats. Using the multiplication rule, determine the probability that a randomly selected member of the 107th

Let event the member selected is a Democrat and event the member selected is a senator

We want . By the multiplication rule,

7. Belief in Aliens

According to the Current Population Reports, 52% of U.S. adults are women. Opinion Dynamics Poll published in USA Today shows that 33% of U.S. women and 54% of U.S. men believe in aliens. What percentage of U.S. adults believe in aliens?

Let event the adult selected believes in aliens and event the adult selected is a man. By law of total probability: ;

8. Lung Disease and Smoking

According to the American Lung Association, 7% of the population has lung disease. Of the people having lung disease, 90% are smokers. Of the people not having lung disease, 20% are smokers. What are the chances that a smoker has lung disease?

Let event the person selected at random is a smoker and event the selected person has lung disease. We want to compute . By Bayes’ Theorem,