Probability

Given: , , and are mutually exclusive events, and the probabilities of these events are given by: - - - We are tasked with finding the set of possible values of such that these probabilities are valid (i.e., they are between 0 and 1). Step 1: Conditions for valid probabilities. For any probability , it must satisfy: $ : \) from this inequality is . 2. For : \) from this inequality is . 3. For : \) from this inequality is . Step 3: Find the intersection of the three ranges. From the three conditions, the valid range for must satisfy all three inequalities. The ranges are: - - - The intersection of these three ranges is: \) $ Conclusion: The set of possible values of is . Thus, the correct answer is: [1/3, 1/2]

Given: We are tasked with finding the probability that a randomly chosen 2x2 determinant with elements 0 and 1 only is non-zero. Step 1: Understand the possible 2x2 matrices. A 2x2 matrix has 4 elements. Since each element can either be 0 or 1, there are a total of possible matrices. The general form of a 2x2 matrix is: $ . Step 2: Calculate the determinant of the matrix. The determinant of a 2x2 matrix is given by: \) . Step 3: Analyze when the determinant is non-zero. 1. **Case 1: and :** - The determinant is . 2. **Case 2: and :** - The determinant is , which is if and if , so it can be non-zero. 3. **Case 3: and :** - The determinant is , which is if and if , so it can be non-zero. 4. **Case 4: and :** - The determinant is , which is if and non-zero if . Step 4: Count the number of matrices with non-zero determinant. Now, let's count how many matrices result in a non-zero determinant: - For and , the determinant is always 0 (1 case). - For and , the determinant is non-zero if and , or and (2 cases). - For and , the determinant is non-zero if (1 case). - For and , the determinant is non-zero if (2 cases). Thus, the total number of matrices with a non-zero determinant is . Step 5: Calculate the probability. Since there are 16 total possible matrices, the probability of choosing a matrix with a non-zero determinant is: \) $ Conclusion: The probability that the determinant chosen is non-zero is .