Understanding the Problem
To find the probability that the difference between two randomly chosen numbers, x and y, from the interval (1, 4) is less than 2, we first define our sample space.
Sample Space
- The total area of the square representing all possible pairs (x, y) is calculated as:
- Length of one side = 4 - 1 = 3
- Total Area = 3 * 3 = 9
Condition for the Difference
The condition we want to satisfy is |x - y| < 2.="" this="" can="" be="" split="" into="" two="" />
1. x - y < />
2. y - x < 2="" />
This means we need to consider the two lines:
- y = x - 2
- y = x + 2
Graphing the Inequalities
- The line y = x - 2 intersects the square at (1, -1) and (3, 1).
- The line y = x + 2 intersects the square at (1, 3) and (3, 5).
However, we only look within the limits of our square, which is bounded by x and y both ranging from 1 to 4.
Calculating the Area of Interest
- The area where |x - y| < 2="" forms="" a="" band="" around="" the="" line="" y="x" within="" the="" />
- This band can be visualized as a parallelogram, excluding the triangular regions outside the given lines.
Final Area Calculation
- The area of the region where the condition holds is found to be 7.
- The probability is the area of the favorable outcomes divided by the total area:
Probability = Area of Favorable Outcomes / Total Area = 7 / 9.
Conclusion
Thus, the probability that the difference between the two chosen numbers is less than 2 is 7/9.