This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Question 1: A bag contains 4 red balls and 6 blue balls. Two balls are drawn at random without replacement. What is the probability that both balls are red? Step 1: Total number of balls = 4 + 6 = 10. Number of ways to draw 2 balls from 10 = 102 = (10 × 9)/(2 × 1) = 45. Step 2: Number of red balls = 4. Number of ways to draw 2 red balls from 4 = 42 = (4 × 3)/(2 × 1) = 6. Step 3: Probability both red = (4)/(2)102 = (6)/(45) = (2)/(15). (2)/(15) Question 2: A fair die is rolled. What is the probability of getting an even number? Step 1: Total outcomes when rolling a die = 6. Step 2: Even numbers = 2, 4, 6, so 3 favorable outcomes. Step 3: Probability = (3)/(6) = (1)/(2). (1)/(2) Question 3: From a standard deck of 52 cards, one card is drawn at random. What is the probability it is a king or a spade? Step 1: Total cards = 52. Step 2: Kings = 4, spades = 13, king of spades counted twice, so kings or spades = 4 + 13 - 1 = 16. Step 3: Probability = (16)/(52) = (4)/(13). (4)/(13) Question 4: A box contains 5 defective and 15 good items. Two items are drawn with replacement. Find the probability both are good. Step 1: Probability first good = (15)/(20) = (3)/(4). Step 2: Probability second good (with replacement) = (3)/(4). Step 3: Probability both good = (3)/(4) × (3)/(4) = (9)/(16). (9)/(16) Question 5: The probability of rain on Monday is 0.4, Tuesday 0.5, independently. Find probability of rain on both days. Step 1: Events independent. Step 2: P(both) = P(Monday) × P(Tuesday) = 0.4 × 0.5 = 0.2. 0.2