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In a high school having equal number of boy students and girl students. 75% of students study science and 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 the a randomly selected girl student studies commerce (rounded off to 3 decimal places) is __________ bits.
Correct answer is '3.322'. Can you explain this answer?
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In a high school having equal number of boy students and girl students...
Given Information:
- In a high school, the number of boy students is equal to the number of girl students.
- 75% of students study science and the remaining 25% study commerce.
- Commerce students are two times more likely to be boys than science students.
To Find:
The amount of information gained in knowing that a randomly selected girl student studies commerce.
Solution:
Step 1: Determine the Probability of a Student Studying Commerce
Since 75% of students study science, the remaining 25% study commerce. Let's assume the total number of students is 'N'. Therefore, the number of students studying commerce is 0.25N.
Step 2: Determine the Probability of a Commerce Student Being a Boy or Girl
It is given that commerce students are two times more likely to be boys than science students. Let's assume the probability of a science student being a boy is 'p'. Therefore, the probability of a commerce student being a boy is 2p.
Since the total number of students is 'N', the number of boys studying science is 0.75Np, and the number of boys studying commerce is 0.25N(2p) = 0.5Np.
The number of girls studying science is 0.75N(1-p), and the number of girls studying commerce is 0.25N(1-2p).
Step 3: Calculate the Probability of a Girl Student Studying Commerce
We know that the number of girls studying commerce is 0.25N(1-2p). The total number of girls is N/2 since the number of boys and girls is equal.
Therefore, the probability of a randomly selected girl student studying commerce is (0.25N(1-2p))/(N/2) = 0.5(1-2p).
Step 4: Calculate the Information Gain
To calculate the information gained, we need to use the formula for information entropy:
Information Gain = -log2(probability)
Using the probability calculated in Step 3, the information gain is:
Information Gain = -log2(0.5(1-2p))
Step 5: Substitute the Given Answer to Find 'p'
In the question, it is mentioned that the correct answer is 3.322 bits. Let's substitute this value into the information gain equation and solve for 'p'.
3.322 = -log2(0.5(1-2p))
Solving this equation will give us the value of 'p', which represents the probability of a science student being a boy.
Therefore, the amount of information gained in knowing that a randomly selected girl student studies commerce is 3.322 bits.
- In a high school, the number of boy students is equal to the number of girl students.
- 75% of students study science and the remaining 25% study commerce.
- Commerce students are two times more likely to be boys than science students.
To Find:
The amount of information gained in knowing that a randomly selected girl student studies commerce.
Solution:
Step 1: Determine the Probability of a Student Studying Commerce
Since 75% of students study science, the remaining 25% study commerce. Let's assume the total number of students is 'N'. Therefore, the number of students studying commerce is 0.25N.
Step 2: Determine the Probability of a Commerce Student Being a Boy or Girl
It is given that commerce students are two times more likely to be boys than science students. Let's assume the probability of a science student being a boy is 'p'. Therefore, the probability of a commerce student being a boy is 2p.
Since the total number of students is 'N', the number of boys studying science is 0.75Np, and the number of boys studying commerce is 0.25N(2p) = 0.5Np.
The number of girls studying science is 0.75N(1-p), and the number of girls studying commerce is 0.25N(1-2p).
Step 3: Calculate the Probability of a Girl Student Studying Commerce
We know that the number of girls studying commerce is 0.25N(1-2p). The total number of girls is N/2 since the number of boys and girls is equal.
Therefore, the probability of a randomly selected girl student studying commerce is (0.25N(1-2p))/(N/2) = 0.5(1-2p).
Step 4: Calculate the Information Gain
To calculate the information gained, we need to use the formula for information entropy:
Information Gain = -log2(probability)
Using the probability calculated in Step 3, the information gain is:
Information Gain = -log2(0.5(1-2p))
Step 5: Substitute the Given Answer to Find 'p'
In the question, it is mentioned that the correct answer is 3.322 bits. Let's substitute this value into the information gain equation and solve for 'p'.
3.322 = -log2(0.5(1-2p))
Solving this equation will give us the value of 'p', which represents the probability of a science student being a boy.
Therefore, the amount of information gained in knowing that a randomly selected girl student studies commerce is 3.322 bits.
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Community Answer
In a high school having equal number of boy students and girl students...
Method 1
Given :
50% of the total students are boys so probability of boys P(B)= 1/2
50% of the total students are girls so probability of girls P(G)= 1/2
75% students studies science, so probability of student studies science, P(S)=3/4
25% students studies commerce, so probability of student studies commerce, P(C)=1/4
Given commerce students are two times more likely to be a boy than are science students.
Let the probability that a science student is boy i.e., 
From total probability theorem, the probability of a selected student to be a boy is
Hence probability of a selected commerce student is a girl,
Probability of randomly selected girl studies commerce is given as
Hence, information in knowing that a randomly selected girl studies commerce is given as
Hence, the amount of information gained in knowing the a randomly selected girl student studies commerce is 3.32 bits.
Method 2 :
Let total number of students is 100.
Given, there are equal number of boys and girl students.
∴ Total number of boys = 50
Total number of girls = 50
Given commerce students are two times more likely to be a boy than are science students.
Let Probability of science student to be boy = p.
∴ Probability of science students to be girl = 1− p
p is defined as,
∴ Number of boys in science = Bs= 75p
∴ Number of girls in science = 75− 75 p
Number of boys in commerce = 50− 75 p
Number of girls in commerce = 25− (50 − 75 p) = 75 p− 25
∴ Probability that if a randomly selected student out of 100 students is a girl then she studies commerce is,
Hence, Information in knowing that a randomly selected student is girl and studies commerce is
Hence, the amount of information gained in knowing the a randomly selected girl student studies commerce is 3.32 bits.
Question_Type: 4
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In a high school having equal number of boy students and girl students. 75% of students study science and 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 the a randomly selected girl student studies commerce (rounded off to 3 decimal places) is __________ bits.Correct answer is '3.322'. Can you explain this answer?
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