(A) : Increasing the number of observations minimizes random errors.
(R ) : Positive and negative random errors occur with equal probability.

Options

A. Both (A) and (R ) are true and (R ) is the correct explanation of (A)
B. Both (A) and (R ) are true and (R ) is not the correct explanation of (A)
C. (A) is true but (R ) is false
D. (A ) is flase but (r) is true.

Answer

To analyze the given statements (A) and (R), we will break down the reasoning step by step.

### Step 1: Understanding Statement (A)
Statement (A) claims that "Increasing the number of observations minimizes random errors."

- **Explanation**: Random errors are variations in measurements that occur due to unpredictable factors. When we take multiple observations, the effect of these random errors can be averaged out. This means that while individual measurements may be affected by random errors, the average of a large number of observations will tend to be closer to the true value.

### Step 2: Understanding Statement (R)
Statement (R) states that "Positive and negative random errors occur with equal probability."

- **Explanation**: Random errors can be either positive (where the measured value is greater than the true value) or negative (where the measured value is less than the true value). The assertion that these errors occur with equal probability means that over a large number of measurements, we expect to see an equal number of positive and negative deviations from the true value.

### Step 3: Connecting (A) and (R)
Now, we need to connect the two statements to determine their validity.

- **Linking Explanation**: If positive and negative random errors occur with equal probability, then as we increase the number of observations, the positive errors will tend to cancel out the negative errors. This cancellation leads to a more accurate average value, thereby minimizing the overall effect of random errors.

### Step 4: Conclusion
Based on the explanations above:

- Statement (A) is true because increasing the number of observations does minimize random errors.
- Statement (R) is also true as it correctly describes the nature of random errors.

Thus, both statements (A) and (R) are true, and (R) provides a valid reason for (A).

### Final Answer:
Both statements are true, and (R) is the correct explanation for (A).

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