Certainly! Below is a transcription of the image content with an explanation suitable for an educational website:---**Transcription and Explanation of Probability Table and Questions**The accompanying data was obtained from the financial aid office of a certain university.**Data Table:**| | Receiving Financial Aid | Not Receiving Financial Aid | Total ||------------------------------|-------------------------|-----------------------------|-----------|| Undergraduates | 4,329 | 3,276 | 7,605 || Graduates | 931 | 286 | 1,217 || **Total** | 5,260 | 3,562 | 8,822 |**Description:**- The table shows the number of undergraduate and graduate students who are either receiving or not receiving financial aid.- The data is divided into two categories across undergraduate and graduate levels, and within these, further into those receiving and not receiving financial aid.**Problem Statement:**Let A be the event that a student is an undergraduate student, and let B be the event that a student selected at random is receiving financial aid.**Tasks:**(a) Find each of the following probabilities. (Round your answers to three decimal places.)- P(A)- P(B)- P(A ∩ B)- P(A | B)- P(B | A)(b) Are the events A and B independent events?- Yes- No**Instructions:**1. **Calculate the Probabilities:** - $ P(A) $: The probability that a student is an undergraduate. - $ P(B) $: The probability that a student is receiving financial aid. - $ P(A ∩ B) $: The probability that a student is both an undergraduate and receiving financial aid. - $ P(A | B) $: The probability that a student is an undergraduate, given that they are receiving financial aid. - $ P(B | A) $: The probability that a student is receiving financial aid, given that they are an undergraduate.2. **Determine Independence:** - To determine if A and B are independent events, check if $ P(A ∩ B) = P(A) \times P(B) $.Use this information to work through the probability calculations to determine relationships and dependencies in student aid data.---This transcription and explanation provide a clear presentation of the data and tasks for educational purposes

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Description: Certainly! Below is a transcription of the image content with an explanation suitable for an educational website:---**Transcription and Explanation of Probability Table and Questions**The accompanying data was obtained from the financial aid office

Given a data from financial aid office of a university Receiving financial aid Not receiving…

Answered: The accompanying data were obtained…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Certainly! Below is a transcription of the image content with an explanation suitable for an educational website:

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**Transcription and Explanation of Probability Table and Questions**

The accompanying data was obtained from the financial aid office of a certain university.

**Data Table:**

| | Receiving Financial Aid | Not Receiving Financial Aid | Total |
|------------------------------|-------------------------|-----------------------------|-----------|
| Undergraduates | 4,329 | 3,276 | 7,605 |
| Graduates | 931 | 286 | 1,217 |
| **Total** | 5,260 | 3,562 | 8,822 |

**Description:**

- The table shows the number of undergraduate and graduate students who are either receiving or not receiving financial aid.
- The data is divided into two categories across undergraduate and graduate levels, and within these, further into those receiving and not receiving financial aid.

**Problem Statement:**

Let A be the event that a student is an undergraduate student, and let B be the event that a student selected at random is receiving financial aid.

**Tasks:**

(a) Find each of the following probabilities. (Round your answers to three decimal places.)

- P(A)
- P(B)
- P(A ∩ B)
- P(A | B)
- P(B | A)

(b) Are the events A and B independent events?
- Yes
- No

**Instructions:**

1. **Calculate the Probabilities:**
- $ P(A) $: The probability that a student is an undergraduate.
- $ P(B) $: The probability that a student is receiving financial aid.
- $ P(A ∩ B) $: The probability that a student is both an undergraduate and receiving financial aid.
- $ P(A | B) $: The probability that a student is an undergraduate, given that they are receiving financial aid.
- $ P(B | A) $: The probability that a student is receiving financial aid, given that they are an undergraduate.

2. **Determine Independence:**
- To determine if A and B are independent events, check if $ P(A ∩ B) = P(A) \times P(B) $.

Use this information to work through the probability calculations to determine relationships and dependencies in student aid data.

---

This transcription and explanation provide a clear presentation of the data and tasks for educational purposes

Expert Solution

Step 1

Given a data from financial aid office of a university

	Receiving financial aid	Not receiving financial aid	Total
Undergraduates	4226	3896	8122
Graduates	1873	739	2612
Total	6099	4635	10734

A is the event that a student selected at random from this university is an undergraduate student.

B is the event that a student selected at random is receiving financial aid.

Step 2

Consider the formula for probability of an event X as:

$P (X) = \frac{N u m b e r o f f a v o r a b l e o u t c o m e s}{T o t a l n u m b e r o f f a v o r a b l e o u t c o m e s}$

(a) Total number of students in university is 10734. Hence total number of favorable outcomes will be

$N (U) = 10734$

A is the event that a student selected at random from this university is an undergraduate student. Since number of undergraduate students is 8122. Hence number of favorable outcomes is

$N (A) = 8122$

Hence probability of A will be

$\begin{array}{rcl} P (A) & = & \frac{N (A)}{N (U)} \\ = & \frac{8122}{10734} \\ = & 0.75666 \\ \approx & 0.757 \end{array}$

Now B is an event that a student selected at random is receiving financial aid. Since number of students receiving financial aid is 6099. Hence number of favorable outcomes is

$N (B) = 6099$

Hence probability of B will be

$\begin{array}{rcl} P (B) & = & \frac{N (B)}{N (U)} \\ = & \frac{6099}{10734} \\ = & 0.56819 \\ \approx & 0.568 \end{array}$

Now event $A \cap B$ means student selected at random from university is an undergraduate student and receiving financial aid. Since number of undergraduate students receiving financial aid is 4226. Hence number of favorable outcomes is

$N (A \cap B) = 4226$

Hence the probability of $A \cap B$ will be

$\begin{array}{rcl} P (A \cap B) & = & \frac{N (A \cap B)}{N (U)} \\ = & \frac{4226}{10734} \\ = & 0.39370 \\ \approx & 0.394 \end{array}$

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