Question

In a shelf there are three mathematics and two physics books. A student takes a book randomly. If he randomly takes, successively for three time by replacing the book already taken every time, then the mean of the number of mathematics books which is treated as random variable is

• A. \(\frac{3}{2}\)
• B. \(\frac{129}{125}\)
• C. \(\frac{9}{5}\)
• D. \(\frac{174}{125}\)

Hint:

Whenever sampling is done "with replacement", the events are independent, and counting successes directly falls under the Binomial Distribution where the mean is simply \(np\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem asks for the mean of the number of mathematics books selected in \(3\) independent trials with replacement. This scenario perfectly models a binomial distribution.
Step 2: Key Formula or Approach:
For a binomial random variable \(X \sim B(n, p)\), where \(n\) is the number of trials and \(p\) is the probability of success in a single trial, the mean (expected value) is given by: \[ E(X) = np \] Step 3: Detailed Explanation:
The total number of books is \(3 + 2 = 5\).
The probability of picking a mathematics book in any single draw is \(p = \frac{3}{5}\).
Since the books are drawn with replacement, the trials are independent. The number of trials is \(n = 3\).
Using the formula for the mean of a binomial distribution: \[ E(X) = 3 \times \frac{3}{5} = \frac{9}{5} \] Step 4: Final Answer:
The mean of the number of mathematics books drawn is \(\frac{9}{5}\).