To solve this problem, we need to calculate the probability of throwing doublets at least twice in four throws with a pair of dice.

Step 1: Calculate the probability of throwing doublets in a single throw.
When we roll a pair of dice, there are 36 possible outcomes (6 outcomes for the first die multiplied by 6 outcomes for the second die). Out of these 36 outcomes, there are 6 outcomes where doublets are thrown (1-1, 2-2, 3-3, 4-4, 5-5, 6-6). Therefore, the probability of throwing doublets in a single throw is 6/36, which simplifies to 1/6.

Step 2: Calculate the probability of throwing doublets at least twice in four throws.
To calculate this probability, we need to consider three cases:
Case 1: Throwing doublets exactly twice
In this case, we need to calculate the probability of throwing doublets twice and any other outcome twice. The probability of throwing doublets twice is (1/6)^2, and the probability of any other outcome is (5/6)^2. Since we can arrange these outcomes in different ways, we need to multiply these probabilities by the number of arrangements, which is given by the binomial coefficient C(4,2) = 6. Therefore, the probability of throwing doublets exactly twice is 6 * (1/6)^2 * (5/6)^2 = 150/1296.

Case 2: Throwing doublets three times
In this case, we need to calculate the probability of throwing doublets three times and any other outcome once. The probability of throwing doublets three times is (1/6)^3, and the probability of any other outcome is 5/6. Again, we need to multiply these probabilities by the number of arrangements, which is given by the binomial coefficient C(4,3) = 4. Therefore, the probability of throwing doublets three times is 4 * (1/6)^3 * (5/6) = 100/1296.

Case 3: Throwing doublets four times
In this case, we need to calculate the probability of throwing doublets four times, which is (1/6)^4 = 1/1296.

Step 3: Calculate the total probability
To calculate the total probability of throwing doublets at least twice, we need to sum up the probabilities from all three cases:
Total probability = Probability of throwing doublets exactly twice + Probability of throwing doublets three times + Probability of throwing doublets four times
Total probability = 150/1296 + 100/1296 + 1/1296 = 251/1296

Simplifying the fraction, we get 19/144.

Therefore, the correct answer is option C) 19/144.