In a simultaneous throw of two fair dice, what is the probability of getting a doublet, that is, the same number on both dice?

Introduction / Context:
This problem tests your ability to count outcomes when two dice are rolled together. A doublet means both dice show the same number, such as (1,1) or (4,4). Identifying how many such outcomes exist and comparing with the total number of possible outcomes is a core probability skill.

Given Data / Assumptions:

• Two fair six sided dice are thrown simultaneously.
• Each die shows one of the numbers 1, 2, 3, 4, 5, 6.
• All ordered pairs of outcomes are equally likely.
• A doublet means both dice show the same number.

Concept / Approach:
When two dice are rolled, the total number of possible ordered outcomes is: Total outcomes = 6 * 6 = 36. A doublet occurs when the first die and the second die show identical values. We simply count the number of such pairs and then compute the probability as favorable outcomes divided by total outcomes.

Step-by-Step Solution:
Step 1: List all possible doublets: (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). Step 2: Count these doublets. There are 6 distinct doublet outcomes. Step 3: Total number of ordered outcomes when two dice are thrown = 6 * 6 = 36. Step 4: Probability of getting a doublet = number of doublets / total outcomes = 6 / 36. Step 5: Simplify 6 / 36 by dividing numerator and denominator by 6 to get 1 / 6.

Verification / Alternative check:
You can view each possible number on the first die and ask what is the chance the second matches it. No matter what the first die shows, the second die has a 1/6 chance of matching that exact number. Since the first die can be anything, and the second must match it, the overall probability of a doublet is 1/6. This reasoning matches the explicit counting of 6 favorable outcomes out of 36 total outcomes.

Why Other Options Are Wrong:
Option 1/3 would suggest that 12 out of 36 outcomes are doublets, which is too many because there are only 6 equal pairs. Option 5/36 treats only 5 outcomes as favorable, which would ignore one of the six possible equal pairs. Option 2/3 is far too large and implies that doublets are very common, which goes against direct outcome counting. Option 1/12 underestimates the true probability and does not align with any correct combination of favorable and total outcomes here.

Common Pitfalls:
Students sometimes confuse ordered pairs with unordered outcomes, or they mistakenly think that some doublets are more likely than others. Another common issue is miscounting the total sample space by assuming 11 or some other wrong number of outcomes. Always remember that two dice give 6 possibilities for each die, and thus 36 ordered outcome pairs in total. For doublets, simply match equal numbers and count those pairs carefully.

Final Answer:
The probability of getting a doublet when two dice are thrown is 1/6.