There are 32 students in a classroom , in which some are girls and rest boys . One student is chosen at random for being monitor in class . Probability of monitor , being a girl is `(3)/(8)` , If two students selected at random from class for debate competition ,then find the probability of being it a girl and a boy ?

Options

A. `(15)/(62)`
B. `(15)/(31)`
C. `(24)/(4)`
D. `(5)/(8)`

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Determine the number of girls and boys
We know that there are a total of 32 students in the classroom. Let's denote the number of girls as \( x \). Therefore, the number of boys will be \( 32 - x \).

### Step 2: Use the given probability to find \( x \)
The probability of selecting a girl as a monitor is given as \( \frac{3}{8} \). The probability can be expressed as:
\[ P(\text{girl}) = \frac{\text{Number of girls}}{\text{Total number of students}} = \frac{x}{32} \]
Setting this equal to the given probability:
\[ \frac{x}{32} = \frac{3}{8} \]

### Step 3: Solve for \( x \)
To solve for \( x \), we can cross-multiply:
\[ 8x = 3 \times 32 \]
\[ 8x = 96 \]
\[ x = \frac{96}{8} = 12 \]
Thus, the number of girls is \( 12 \).

### Step 4: Calculate the number of boys
Now, we can find the number of boys:
\[ \text{Number of boys} = 32 - x = 32 - 12 = 20 \]

### Step 5: Calculate the probability of selecting one girl and one boy
Next, we need to find the probability of selecting one girl and one boy when two students are chosen at random. The total number of ways to choose 2 students from 32 is given by:
\[ \text{Total ways} = \binom{32}{2} = \frac{32 \times 31}{2} = 496 \]

### Step 6: Calculate the favorable outcomes for one girl and one boy
The number of ways to choose 1 girl from 12 is:
\[ \text{Ways to choose 1 girl} = \binom{12}{1} = 12 \]
The number of ways to choose 1 boy from 20 is:
\[ \text{Ways to choose 1 boy} = \binom{20}{1} = 20 \]
Thus, the total number of favorable outcomes for selecting one girl and one boy is:
\[ \text{Favorable outcomes} = 12 \times 20 = 240 \]

### Step 7: Calculate the probability
Now, we can calculate the probability of selecting one girl and one boy:
\[ P(\text{one girl and one boy}) = \frac{\text{Favorable outcomes}}{\text{Total ways}} = \frac{240}{496} \]
To simplify this fraction, we can divide both the numerator and the denominator by 16:
\[ P(\text{one girl and one boy}) = \frac{15}{31} \]

### Final Answer
Thus, the probability of selecting one girl and one boy is:
\[ \frac{15}{31} \]

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