Random Experiments

To solve this problem, we need to understand the working of a screw gauge and calculate the correct measurement of the wire's radius using given parameters.

1. Given that the pitch of the screw gauge is . This means that one complete turn of the circular scale displaces the screw by .
2. The number of divisions on the circular scale is . Therefore, the least count of the screw gauge is calculated as:

1. The zero error is mentioned as divisions below the reference line. This translates to a value of: .
2. When the wire is placed between the jaws, the first linear scale division is visible, and the circular scale shows the division aligned with the reference line. This indicates a reading from the circular scale of: .
3. Therefore, the total linear reading is: (since the first linear scale division is visible, we add ).
4. To find the corrected diameter of the wire, we subtract the zero error from the total linear reading: .
5. The radius of the wire is half of the corrected diameter, therefore: .

Hence, the radius of the wire is 0.82 mm, which matches the correct answer option.

We are given that there are 3 rotten apples and 7 good apples. 4 apples are drawn randomly without replacement. We need to find the value of , where is the mean and is the variance of the random variable , which represents the number of rotten apples drawn.

Step 1: Defining the Problem.
The number of rotten apples drawn, , follows a hypergeometric distribution, since we are drawing without replacement. The probability mass function (PMF) of for a hypergeometric distribution is given by: where: - is the number of rotten apples,
- is the number of good apples,
- is the total number of apples drawn,
- is the number of rotten apples drawn.

Step 2: Mean and Variance of Hypergeometric Distribution.
For a hypergeometric distribution, the mean and variance are given by the formulas: where: - (number of draws),
- (total number of rotten apples),
- (total number of apples).
Calculating the Mean :
Calculating the Variance :

Step 3: Calculating . Now, calculate :

Thus, the value of is , and the correct answer is option (4).