Final answer:
To find the probability of event A, defined as 'X₁ < X₂', where X₁ is the result of the first roll and X₂ is the result of the second roll of a fair die, we need to determine the number of favorable outcomes and the total number of possible outcomes. The probability of event A occurring is 15/36 or 5/12.
Step-by-step explanation:
To find the probability of event A, defined as 'X₁ < X₂', where X₁ is the result of the first roll and X₂ is the result of the second roll of a fair die, we need to determine the number of favorable outcomes and the total number of possible outcomes.
For each roll, there are 6 possible outcomes since the die has 6 sides numbered from 1 to 6. So, the total number of possible outcomes for the two rolls is 6 × 6 = 36.
To calculate the number of favorable outcomes, we need to count the number of times X₁ is less than X₂. In this case, the favorable outcomes are:
{(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 5), (4, 6), (5, 6)}
There are 15 favorable outcomes. Therefore, the probability of event A occurring is 15/36 or 5/12.