To find the probability of exactly one odd number turning up among the three outcomes when a die is rolled three times, we can use the concept of probability and counting.

Let's break down the problem step by step:

Step 1: Determine the total number of possible outcomes.
When a die is rolled, there are six possible outcomes, which are the numbers 1, 2, 3, 4, 5, and 6.

Step 2: Determine the total number of favorable outcomes.
To find the number of favorable outcomes, we need to consider the different cases where exactly one odd number turns up among the three outcomes.

Case 1: The odd number appears in the first roll.
In this case, the odd number can be any one of the three odd numbers (1, 3, or 5). The remaining two rolls can be any of the six numbers on the die. Therefore, the total number of outcomes for this case is 3 * 6 * 6 = 108.

Case 2: The odd number appears in the second roll.
Similar to Case 1, the odd number can be any one of the three odd numbers. The first roll and the third roll can be any of the six numbers on the die. Hence, the total number of outcomes for this case is 6 * 3 * 6 = 108.

Case 3: The odd number appears in the third roll.
Again, the odd number can be any one of the three odd numbers. The first and second rolls can be any of the six numbers on the die. Therefore, the total number of outcomes for this case is 6 * 6 * 3 = 108.

Step 3: Calculate the total number of favorable outcomes.
To find the total number of favorable outcomes, we need to sum up the outcomes from all three cases.
Total favorable outcomes = 108 + 108 + 108 = 324

Step 4: Calculate the probability.
The probability of an event is given by the ratio of favorable outcomes to total outcomes.
Probability = Total favorable outcomes / Total possible outcomes
Probability = 324 / 6^3 = 324 / 216 = 3/8

Therefore, the correct answer is option 'B' (3/8).