A class consists of 80 students 25 of them are girls and 55 boys. If 10 of them are rich and the remaining poor and also 20 of them are intelligent them the probability of selecting an intelligent rich girl is

Options

A. `5//128`
B. `25//128`
C. `5//512`
D. none

Answer

To solve the problem of finding the probability of selecting an intelligent rich girl from a class of 80 students, we can follow these steps:

### Step 1: Identify the Total Number of Students
The total number of students in the class is given as 80.

### Step 2: Identify the Number of Intelligent Students
We are told that there are 20 intelligent students in the class.

### Step 3: Identify the Number of Rich Students
According to the problem, there are 10 rich students in the class.

### Step 4: Identify the Number of Girls
The total number of girls in the class is 25.

### Step 5: Calculate the Probability of Selecting an Intelligent Student
The probability of selecting an intelligent student is given by the formula:
\[ P(\text{Intelligent}) = \frac{\text{Number of Intelligent Students}}{\text{Total Students}} = \frac{20}{80} = \frac{1}{4} \]

### Step 6: Calculate the Probability of Selecting a Rich Student
The probability of selecting a rich student is given by:
\[ P(\text{Rich}) = \frac{\text{Number of Rich Students}}{\text{Total Students}} = \frac{10}{80} = \frac{1}{8} \]

### Step 7: Calculate the Probability of Selecting a Girl
The probability of selecting a girl is given by:
\[ P(\text{Girl}) = \frac{\text{Number of Girls}}{\text{Total Students}} = \frac{25}{80} = \frac{5}{16} \]

### Step 8: Combine the Probabilities
To find the probability of selecting an intelligent rich girl, we multiply the probabilities of the three independent events:
\[ P(\text{Intelligent Rich Girl}) = P(\text{Intelligent}) \times P(\text{Rich}) \times P(\text{Girl}) \]

Substituting the values we calculated:
\[ P(\text{Intelligent Rich Girl}) = \left(\frac{20}{80}\right) \times \left(\frac{10}{80}\right) \times \left(\frac{25}{80}\right) \]

### Step 9: Simplify the Expression
Calculating the above expression:
\[ P(\text{Intelligent Rich Girl}) = \frac{20}{80} \times \frac{10}{80} \times \frac{25}{80} = \frac{1}{4} \times \frac{1}{8} \times \frac{5}{16} \]
\[ = \frac{1 \times 1 \times 5}{4 \times 8 \times 16} = \frac{5}{512} \]

### Final Answer
Thus, the probability of selecting an intelligent rich girl is:
\[ \frac{5}{512} \]

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