Answered

4. In a school, there are 542 students, of whom 282 are girls. Of the 542 students, 364 walk to school, and 153 of those are girls. Find the probability that a student chosen at random:

a) is a boy
b) is a boy who does not walk to school
c) does not walk to school, given that they are a boy
d) is a girl, given that they walk to school.



Answer :

GinnyAnswer
Certainly! Let's solve the given problem step-by-step:

### Given Information:
- Total students: 542
- Total girls: 282
- Total students who walk to school: 364
- Girls who walk to school: 153

### Step-by-Step Solution:

#### 1. Calculate the number of boys:
[tex]\[ \text{Total boys} = \text{Total students} - \text{Total girls} \][/tex]
[tex]\[ = 542 - 282 = 260 \][/tex]

#### 2. Calculate the number of boys who walk to school:
[tex]\[ \text{Total boys who walk} = \text{Total students who walk} - \text{Girls who walk} \][/tex]
[tex]\[ = 364 - 153 = 211 \][/tex]

#### 3. Calculate the number of boys who do not walk to school:
[tex]\[ \text{Total boys who do not walk} = \text{Total boys} - \text{Boys who walk} \][/tex]
[tex]\[ = 260 - 211 = 49 \][/tex]

### a) Probability that a student chosen at random is a boy:

[tex]\[ P(\text{Boy}) = \frac{\text{Total boys}}{\text{Total students}} \][/tex]
[tex]\[ = \frac{260}{542} \approx 0.4797 \][/tex]

### b) Probability that a student chosen at random is a boy who does not walk to school:

[tex]\[ P(\text{Boy who does not walk}) = \frac{\text{Boys who do not walk}}{\text{Total students}} \][/tex]
[tex]\[ = \frac{49}{542} \approx 0.0904 \][/tex]

### c) Probability that a student does not walk to school, given that they are a boy:

[tex]\[ P(\text{Does not walk} \mid \text{Boy}) = \frac{\text{Boys who do not walk}}{\text{Total boys}} \][/tex]
[tex]\[ = \frac{49}{260} \approx 0.1885 \][/tex]

### d) Probability that a student is a girl, given that they walk to school:

[tex]\[ P(\text{Girl} \mid \text{Walk}) = \frac{\text{Girls who walk}}{\text{Total students who walk}} \][/tex]
[tex]\[ = \frac{153}{364} \approx 0.4203 \][/tex]

### Summary:
- The probability that a student chosen at random is a boy: [tex]\(\approx 0.4797\)[/tex]
- The probability that a student chosen at random is a boy who does not walk to school: [tex]\(\approx 0.0904\)[/tex]
- The probability that a student does not walk to school, given that they are a boy: [tex]\(\approx 0.1885\)[/tex]
- The probability that a student is a girl, given that they walk to school: [tex]\(\approx 0.4203\)[/tex]