Mathematics of Chance

Probability is always between 0 and 1, Probability of an impossible event is 0, Probability of a certain event is 1, Sum of all probabilities in a sample space is 1

Step 1: Probability range is 0 ≤ P(E) ≤ 1 always. Step 2: Impossible events have P = 0. Step 3: Certain events have P = 1. Step 4: Probability cannot exceed 1. Step 5: All probabilities in a sample space sum to 1.

1/2

Step 1: List all two-digit numbers: 12, 13, 14, 21, 23, 24, 31, 32, 34, 41, 42, 43 = 12 numbers. Step 2: Count even numbers (ending in 2 or 4): 12, 14, 32, 34, 42, 24 = 6 numbers. Step 3: Probability = 6/12 = 1/2.

1/2

Step 1: The rectangle is divided into two equal triangles. Step 2: Each triangle has area = (1/2) × rectangle area. Step 3: Probability = Area of one triangle / Area of rectangle = (1/2 × area) / area = 1/2.

Sum = 2, Sum = 7, Sum = 12

Step 1: Minimum sum = 1+1 = 2. Step 2: Maximum sum = 6+6 = 12. Step 3: Sum = 7 is possible (like 3+4). Step 4: Sum = 13 is impossible (exceeds maximum). Step 5: Sum = 1 is impossible (less than minimum).