Final answer:
The probability that the first die roll is less than the second on a fair six-sided die is 5/12. This calculation assumes the die is fair and the rolls are independent. The law of large numbers tells us that over many trials, the observed probability will tend to match the theoretical probability.
Step-by-step explanation:
The student is asking about the probability of one outcome being less than another when rolling a fair six-sided die twice. To calculate this, we must consider all the possible rolls where the first roll is less than the second roll. For the first roll of 1, there are five possibilities for the second roll (2, 3, 4, 5, 6). For the first roll of 2, there are four possibilities for the second roll (3, 4, 5, 6), and so on, until the first roll of 5, where there is only one possibility for the second roll (6).
Summing these up, we have 5+4+3+2+1 = 15 successful outcomes. Since there are 6 possible outcomes for each roll, there are a total of 6 * 6 = 36 possible combinations when rolling the die twice. Therefore, the probability is the number of successful outcomes divided by the number of possible outcomes: P(success) = 15/36, which simplifies to 5/12 after dividing numerator and denominator by 3.
The law of large numbers states that as the number of trials increases, the experimental probability will approach the theoretical probability. Therefore, while small numbers of trials might not show a probability close to 5/12, over many trials, the result will tend to get closer to this theoretical value.