## Question

a The table below provides summary information about students in a class. The gender of each student and the major is given. If a student is randomly selected from this group Male Female 45 60 Accounting Economics 30 55 Business 60 100

STATISTICS PROBABILITY

a) determine the number of students in a class.

b) what is the probability that the student is male?

c) what is the probability that the student is female and major in Business?

d) what is the probability that the student is female or major in Accounting?

e) if the student is majoring in Economics, what is the probability that the student is a male?

f) what is the probability that the students not majoring in Economics?

g) determine whether the event that the student is male and major in Economics is an independent event. Prove it.

## Answer

**a) Determine the number of students in a class.**

To determine the number of students in the class, we need to add the number of students in each category.

```
Number of students = Male + Female
Number of students = 45 + 60 = 105
```

There are 105 students in the class.

**b) What is the probability that the student is male?**

To calculate the probability that the student is male, we need to divide the number of male students by the total number of students.

```
Probability (male) = Male / Total students
Probability (male) = 45 / 105 = 0.43
```

The probability that the student is male is 0.43.

**c) What is the probability that the student is female and major in Business?**

To calculate the probability that the student is female and major in Business, we need to divide the number of female students majoring in Business by the total number of students.

```
Probability (female and Business) = Female and Business / Total students
Probability (female and Business) = 60 / 105 = 0.57
```

The probability that the student is female and major in Business is 0.57.

**d) What is the probability that the student is female or major in Accounting?**

To calculate the probability that the student is female or major in Accounting, we need to add the probability that the student is female to the probability that the student is majoring in Accounting.

Probability (female or Accounting) = Probability (female) + Probability (Accounting) – Probability (female and Accounting) Probability (female or Accounting) = 0.57 + 0.30 – 0.07 = 0.80

The probability that the student is female or major in Accounting is 0.80.

**e) If the student is majoring in Economics, what is the probability that the student is a male?**

To calculate the conditional probability that the student is male given that the student is majoring in Economics, we need to divide the number of male students majoring in Economics by the total number of students majoring in Economics.

```
Probability (male | Economics) = Male and Economics / Total Economics
Probability (male | Economics) = 30 / 85 = 0.35
```

The probability that the student is a male given that the student is majoring in Economics is 0.35.

**f) What is the probability that the students not majoring in Economics?**

To calculate the probability that the student is not majoring in Economics, we need to subtract the probability that the student is majoring in Economics from 1.

```
Probability (not Economics) = 1 - Probability (Economics)
Probability (not Economics) = 1 - 0.80 = 0.20
```

The probability that the student is not majoring in Economics is 0.20.

**g) Determine whether the event that the student is male and major in Economics is an independent event. Prove it.**

Two events are independent if the occurrence of one event does not affect the probability of the other event. To determine whether the event that the student is male and major in Economics is independent, we need to compare the probability of both events happening to the product of the probability of one event happening and the probability of the other event happening.

```
P(male and Economics) = 0.30
P(male) = 0.43
P(Economics) = 0.80
```

If the events are independent, then:

```
P(male and Economics) = P(male) * P(Economics)
0.30 = 0.43 * 0.80
0.30 ≠ 0.344
```

Since the probability of both events happening is not equal to the product of the probability of one event happening and the probability of the other event happening, the events are not independent.

Therefore, the event that the student is male and major in Economics is not an independent event.