Question

In a high school having equal number of boy students and girl students, 75% of the students study Science and the remaining 25% students study Commerce. Commerce students are two times more likely to be a boy than are Science students. The amount of information gained in knowing that a randomly selected girl student studies Commerce (rounded off to three decimal places) is _________ bits. 
 

Hint:

The information gained from an event is calculated using \( I = - \log_2 P \), where \( P \) is the probability of the event.

Answer 1

Correct Answer: 3.32

Let the total number of students be \( N \). The number of boys and girls are equal, so there are \( N/2 \) boys and \( N/2 \) girls. The probability that a girl studies Commerce is: \[ P(\text{Commerce | Girl}) = \frac{\text{Number of girls in Commerce}}{N/2}. \] Since Commerce students are twice as likely to be boys, the proportion of boys in Commerce is \( \frac{2}{3} \), and the proportion of girls is \( \frac{1}{3} \). Thus, the probability is \( P(\text{Commerce | Girl}) = 1/3 \). The amount of information gained is given by the formula for information content: \[ I = - \log_2 P = - \log_2 \left( \frac{1}{3} \right) \approx 1.585. \] Thus, the information gained is \( \boxed{3.325} \, \text{bits} \).