Mathematics of Chance

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Probability is a measure of how likely an event is to occur. It is calculated as: Probability = Number of favorable outcomes / Total number of possible outcomes Example: In a box with 6 black balls …

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Step-by-step solution: Step 1: Count total balls = 8 red + 12 blue = 20 balls Step 2: Count favorable outcomes = 8 red balls Step 3: Apply probability formula Probability of red ball = 8/20 = 2/5 = …

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Step-by-step solution: Step 1: Identify multiples of 5 from 1 to 50 Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 Count = 10 numbers Step 2: Total possible outcomes = 50 Step 3: Calculate p…

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In geometric probability: Probability = Area of favorable region / Area of total region Example: A triangle drawn inside a rectangle has half the area of the rectangle. If a point is marked randoml…

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Step-by-step solution: Let circle radius = r Step 1: Area of circle = πr² Step 2: For inscribed square, diagonal = 2r Side of square = 2r/√2 = r√2 Area of square = (r√2)² = 2r² Step 3: Calculate pr…

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Step-by-step solution: Step 1: Count balls in each box Box A: 3 + 2 = 5 balls total Box B: 4 + 1 = 5 balls total Step 2: Apply multiplication principle Total outcomes = Outcomes from Box A × Outcome…

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Step-by-step solution: From FC6: Total outcomes = 25 Step 1: Count favorable outcomes Red balls in Box A = 3 Red balls in Box B = 4 Both red outcomes = 3 × 4 = 12 Step 2: Calculate probability P(bo…

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'At least one' means one or more events occur. Two methods to calculate: Method 1 (Direct): Count all cases where condition is satisfied Method 2 (Complement): P(at least one) = 1 - P(none) This is…