1.Among 90 pieces of mail delivered to the library of a college,50 are addressed to the faculty of Business Administration and 40 are addressed to the Communication faculty.if two of these pieces of mail are delivered to the Dean's office by mistake,and the selection is random,what are the probabilities that:

I.Both of them should have been delivered to the faculty of Business Administration.
II.both of them should have been delivered to the faculty of Arts.
III.One should have been delivered to the faculty of Arts and other to the Faculty of Business Administration.

2.In a statistics class,there are 18 juniors and 10 seniors;6 of the seniors are females and 12 of the juniors are males.if a student is selected at random,find the probability of selecting
I.a junior or a female.
II.a senior or a female.
III.a junior or a senior.

1. If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.

I. (50/90)^2 = ?
II. Generalize from I.
III. (50/90)(40/90) = ?

2. Either-or probabilities are found by adding the individual probabilities.

III. 18/28 + 10/28 = ?

I. Both of them should have been delivered to the faculty of Business Administration:

To find the probability, we need to look at the number of ways both pieces of mail can be selected from the mail delivered to the Dean's office and compare it to the total number of ways the pieces of mail can be selected.

Total pieces of mail delivered to the Dean's office = 2
Total pieces of mail addressed to the faculty of Business Administration = 50

The probability can be calculated using the formula:
Probability = (Number of favorable outcomes)/(Total number of outcomes)

Number of favorable outcomes = 2 (both selected from the 50 pieces addressed to the Business Administration faculty)
Total number of outcomes = 90 (total pieces of mail delivered to the library)

Probability = 2/90 = 1/45

So, the probability that both pieces of mail should have been delivered to the faculty of Business Administration is 1/45.

II. Both of them should have been delivered to the faculty of Arts:
Similarly, the probability of both pieces of mail being selected from the 40 pieces addressed to the Communication faculty is:
Probability = 2/90 = 1/45

So, the probability that both pieces of mail should have been delivered to the faculty of Arts is 1/45.

III. One should have been delivered to the faculty of Arts and the other to the Faculty of Business Administration:
To find the probability, we need to consider the number of ways one piece of mail can be selected from the 40 pieces addressed to the Communication faculty and the number of ways the other piece can be selected from the 50 pieces addressed to the Business Administration faculty. We multiply these numbers to get the total number of favorable outcomes.

Number of favorable outcomes = (40 * 50) = 2000 (select 1 from the Communication faculty and 1 from the Business Administration faculty)
Total number of outcomes = 90 (total pieces of mail delivered to the library)

Probability = 2000/90 = 20/9

So, the probability that one piece of mail should have been delivered to the faculty of Arts and the other to the Faculty of Business Administration is 20/9.

Moving on to the second question...

I. Probability of selecting a junior or a female:
Number of juniors = 18
Number of females = 6

Total number of juniors + total number of females = 18 + 6 = 24

Probability = (Number of favorable outcomes)/(Total number of outcomes)
Number of favorable outcomes = 18 (selecting a junior) + 6 (selecting a female) - 0 (no overlap)
Total number of outcomes = 18 + 10 (total number of juniors and seniors)

Probability = 24/28 = 6/7

So, the probability of selecting a junior or a female is 6/7.

II. Probability of selecting a senior or a female:
Number of seniors = 10

Total number of seniors + total number of females = 10 + 6 = 16

Probability = (Number of favorable outcomes)/(Total number of outcomes)
Number of favorable outcomes = 10 (selecting a senior) + 6 (selecting a female) - 0 (no overlap)
Total number of outcomes = 18 + 10 (total number of juniors and seniors)

Probability = 16/28 = 4/7

So, the probability of selecting a senior or a female is 4/7.

III. Probability of selecting a junior or a senior:
Probability = (Number of favorable outcomes)/(Total number of outcomes)
Number of favorable outcomes = 18 (selecting a junior) + 10 (selecting a senior) - 0 (no overlap)
Total number of outcomes = 18 + 10 (total number of juniors and seniors)

Probability = 28/28 = 1

So, the probability of selecting a junior or a senior is 1.

To solve these probability questions, we need to use the concept of combinations. Let's go step-by-step to answer each question.

1. Probabilities with mail delivery:
I. The probability that both pieces of mail should have been delivered to the faculty of Business Administration can be calculated as follows:
- Total pieces of mail addressed to the Business Administration faculty = 50
- Total mail delivered to the Dean's office by mistake = 2
So, the probability is (50/90) * (49/89), as we are selecting two pieces of mail without replacement.
Simplifying this expression gives us the answer.

II. The probability that both pieces of mail should have been delivered to the faculty of Arts:
- Total pieces of mail addressed to the Arts faculty = 40
- Total mail delivered to the Dean's office by mistake = 2
So, the probability is (40/90) * (39/89), as we are selecting two pieces of mail without replacement.
Simplify this expression to get the answer.

III. The probability that one piece of mail should have been delivered to the faculty of Arts and the other to the faculty of Business Administration:
Here, we calculate the probability of each piece of mail being delivered to the respective faculty and then multiply them together. Since there are two ways to arrange these two mails (one to Arts and one to Business Administration), we multiply the resulting probability by 2.
- Probability of a piece of mail being delivered to the Arts faculty = 40/90
- Probability of a piece of mail being delivered to the Business Administration faculty = 50/89 (since one mail is already removed from the Business Administration faculty)
Multiply these probabilities together and multiply the result by 2 to account for the two possible arrangements.

2. Probabilities in the statistics class:
I. The probability of selecting a junior or a female:
- Total juniors = 18
- Total females (seniors + juniors) = 6 + 12 = 18
- Total students = 18 + 10 = 28 (juniors + seniors)
Add the probabilities of selecting a junior and selecting a female and subtract the probability of selecting a junior female (to avoid double-counting).

II. The probability of selecting a senior or a female:
- Total seniors = 10
- Total females = 6
Add the probabilities of selecting a senior and selecting a female and subtract the probability of selecting a senior female (to avoid double-counting).

III. The probability of selecting a junior or a senior:
- Total juniors = 18
- Total seniors = 10
Total students = 28 (juniors + seniors)
Calculate the probability of selecting a junior and a senior and add them together.

I hope these step-by-step explanations help you solve the probability questions! Let me know if you need further assistance.

To solve these probability questions, we will need to use the concept of probability and apply it to the given information.

1. Probability of mail delivery:
Let's find the total number of pieces of mail delivered to the library, which is 90.

I. Both pieces of mail should have been delivered to the faculty of Business Administration:
There are 50 mail items addressed to the faculty of Business Administration. Since you are selecting two mail pieces by random, the probability of the first mail being chosen correctly is 50/90. After the first mail is taken, there are 49 mail items addressed to the faculty of Business Administration left out of a total of 89 mail pieces remaining. Therefore, the probability of the second mail item being chosen correctly is 49/89.
To find the probability of both events happening, we multiply the probabilities: (50/90) * (49/89) = 2450/8010 = 0.3054 (rounded to 4 decimal places) or 30.54% (rounded to 2 decimal places).

II. Both pieces of mail should have been delivered to the faculty of Arts:
There are 40 mail items addressed to the Communication faculty. Following the same reasoning as in the first question, we can find the probability of both mails being chosen correctly: (40/90) * (39/89) = 40/1989 ≈ 0.0201 (rounded to 4 decimal places) or 2.01% (rounded to 2 decimal places).

III. One piece should have been delivered to the faculty of Arts and the other to the faculty of Business Administration:
We have the same probabilities as mentioned above. Since there are two possible ways this can happen (faculty of Arts first, then Business Administration or vice versa), we calculate the probability for each case and sum them:
Case 1: (40/90) * (50/89) ≈ 0.2472 (rounded to 4 decimal places) or 24.72% (rounded to 2 decimal places).
Case 2: (50/90) * (40/89) ≈ 0.2472 (rounded to 4 decimal places) or 24.72% (rounded to 2 decimal places).
Total probability: 0.2472 + 0.2472 = 0.4944 (rounded to 4 decimal places) or 49.44% (rounded to 2 decimal places).

2. Probability of selecting students:
Let's find the total number of students in the statistics class, which is the sum of juniors and seniors: 18 + 10 = 28.

I. Probability of selecting a junior or a female:
There are 18 juniors in the class, and 6 of the seniors are females. We want to find the probability of selecting a junior or a female, which means we need to sum the probabilities:
Probability of selecting a junior: 18/28.
Probability of selecting a female senior: 6/28.
Total probability: 18/28 + 6/28 = 24/28 = 0.8571 (rounded to 4 decimal places) or 85.71% (rounded to 2 decimal places).

II. Probability of selecting a senior or a female:
We will use a similar approach here.
Probability of selecting a senior: 10/28.
Probability of selecting a female: 6/28.
Total probability: 10/28 + 6/28 = 16/28 = 0.5714 (rounded to 4 decimal places) or 57.14% (rounded to 2 decimal places).

III. Probability of selecting a junior or a senior:
Probability of selecting a junior: 18/28.
Probability of selecting a senior: 10/28.
Total probability: 18/28 + 10/28 = 28/28 = 1 (rounded to 2 decimal places) or 100%.

Remember, probability calculations require taking into account the total number of possible outcomes and the number of favorable outcomes.