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question:An opaque bag contains 2 yellow balls and 3 red balls. These balls are identical except for their colors. After mixing them up, the probability of drawing a red ball is ______.
answer:To solve this problem, we start by understanding the total number of balls in the opaque bag, which consists of 2 yellow balls and 3 red balls. The balls are identical in every aspect except for their colors. Given this, the total number of balls in the bag is the sum of the yellow balls and the red balls, which is 2 + 3 = 5 balls. When we mix them up and draw one ball randomly, the probability of drawing a red ball is calculated by dividing the number of red balls by the total number of balls. Therefore, the probability is: frac{text{Number of red balls}}{text{Total number of balls}} = frac{3}{2+3} = frac{3}{5} Thus, the probability of drawing a red ball from the bag after mixing them up is boxed{frac{3}{5}}.
question:Given the universal set U={1, 2, 4} and the set A={1, 4}, find the complement of A with respect to U, denoted as complement_U A.
answer:The complement of a set A with respect to the universal set U is defined as the set of all elements in U that are not in A. In this case, since U={-1, 2, 4} and A={-1, 4}, the elements of U not in A are those that are in U but not in A. To find complement_U A: 1. List all elements of U: U={-1, 2, 4}. 2. Identify the elements of A: A={-1, 4}. 3. Exclude any elements from U that are in A. 4. The resulting set is the complement of A with respect to U. After excluding the elements of A from U, we are left with a single element, {2}. Thus, the complement of A with respect to U is: complement_U A = {2} Therefore, the answer is boxed{{2}}.
question:a 4 times 4 square, a piece is initially placed in a cell marked in gray. Each move, the piece moves to a neighboring cell along the direction indicated by the arrow in that cell. Additionally, after each move, the arrow in the cell the piece was just in is reversed. From which cell will the piece exit the boundary of the square? Indicate the row and column of that cell in your answer.
answer:1. **Initial Position:** The problem states that we have a chip on a marked cell in a 4 times 4 grid. We aim to determine from which cell the chip will exit the grid after following the specified rules. 2. **Movement Rules:** - The chip moves to an adjacent cell in the direction indicated by the arrow on the current cell. - After each move, the arrow in the cell where the chip just was changes to the opposite direction. 3. **Analyzing the Path:** Let's follow the path of the chip as per the arrows and apply the direction change rule. 4. **Step-by-Step Calculation:** - **Starting position:** Cell C2 (assuming this is the marked cell). - Move to C3. - Move to B3. - Move to A3. - Move to A4. - Move to B4. - Move to B3. - Move to C3. - Move to D3. - Move to D2. - Move to C2. - Move to C1. - Move to B1. - Move to A1. - Move to A2. 5. **Exiting the Grid:** Based on the path traced, we can observe that the chip will eventually move to cell A2 (row A, column 2) and from there it will make its move that will take it outside the grid. 6. **Conclusion:** The chip will exit the grid from the cell located in row A and column 2. [ boxed{A text{ row, 2 column}} ]
question:Evaluate leftlfloor frac{7}{2} rightrfloor.
answer:The fraction -frac{7}{2} represents the decimal -3.5. The floor function leftlfloor x rightrfloor of a number x is defined as the greatest integer less than or equal to x. - Since -3.5 is between -4 and -3, the greatest integer less than or equal to -3.5 is -4. Therefore, leftlfloor -frac{7}{2} rightrfloor = boxed{-4}.