Probability Concepts

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Probability Concepts · Grade 7 · Worksheet 3

1. P(rolling a 1 or 5 on a fair six-sided die) = ? Answer: 1/3 Solution: A fair six-sided die has outcomes: 1, 2, 3, 4, 5, 6. So, total possible outcomes = 6. Favorable outcomes for rolling a 1 or 5 are: 1, 5.
   Full step-by-step solution

   Step 1: A fair six-sided die has outcomes: 1, 2, 3, 4, 5, 6. So, total possible outcomes = 6. Step 2: Favorable outcomes for rolling a 1 or 5 are: 1, 5. So, number of favorable outcomes = 2. Step 3: Probability = (Number of favorable outcomes) / (Total possible outcomes) = 2/6. Step 4: Simplify the fraction: 2/6 = 1/3. The answer is 1/3.

2. P(rolling a number divisible by 3 on a fair twelve-sided die) = ? Answer: 1/3 Solution: A fair twelve-sided die has numbers 1 through 12, so there are 12 total possible outcomes. Numbers divisible by 3 between 1 and 12 are: 3, 6, 9, 12. That's 4 favorable outcomes.
   Full step-by-step solution

   Step 1: A fair twelve-sided die has numbers 1 through 12, so there are 12 total possible outcomes. Step 2: Numbers divisible by 3 between 1 and 12 are: 3, 6, 9, 12. That's 4 favorable outcomes. Step 3: Probability = (favorable outcomes) / (total outcomes) = 4/12 Step 4: Simplify the fraction: 4/12 = 1/3 The answer is 1/3.

3. P(drawing a card numbered with a multiple of 7 from a bag containing cards numbered 7, 21, 35, 49, 63, 77, 91, 105, 119, 133, 147, 161, 175, 189, 203) = ? Answer: 1 Solution: Count the total number of cards in the bag. The cards are numbered: 7, 21, 35, 49, 63, 77, 91, 105, 119, 133, 147, 161, 175, 189, 203. That is 15 cards total.
   Full step-by-step solution

   Step 1: Count the total number of cards in the bag. The cards are numbered: 7, 21, 35, 49, 63, 77, 91, 105, 119, 133, 147, 161, 175, 189, 203. That is 15 cards total. Step 2: Identify the favorable outcomes. The problem asks for drawing a card with a number that is a multiple of 7. All numbers listed are multiples of 7 (7×1, 7×3, 7×5, 7×7, 7×9, 7×11, 7×13, 7×15, 7×17, 7×19, 7×21, 7×23, 7×25, 7×27, 7×29). So there are 15 favorable outcomes. Step 3: Calculate the probability. Probability = (Number of favorable outcomes) / (Total number of outcomes) = 15/15. Step 4: Simplify the fraction. 15/15 = 1. The answer is 1.

4. Emma is playing a board game that uses two standard six-sided dice. She needs to roll a sum of 8 or more to pass her turn. What is the probability that Emma will successfully pass her turn on her next roll? Express your answer as a simplified fraction. Answer: 5/12 Solution: Find the total number of possible outcomes when rolling two dice. Each die has 6 faces, so total outcomes = 6 × 6 = 36. Identify the favorable outcomes where the sum is 8 or more.
   Full step-by-step solution

   Step 1: Find the total number of possible outcomes when rolling two dice. Each die has 6 faces, so total outcomes = 6 × 6 = 36. Step 2: Identify the favorable outcomes where the sum is 8 or more. Step 3: List all dice combinations with sum ≥ 8: - Sum of 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes - Sum of 9: (3,6), (4,5), (5,4), (6,3) → 4 outcomes - Sum of 10: (4,6), (5,5), (6,4) → 3 outcomes - Sum of 11: (5,6), (6,5) → 2 outcomes - Sum of 12: (6,6) → 1 outcome Step 4: Count all favorable outcomes: 5 + 4 + 3 + 2 + 1 = 15 Step 5: Calculate probability: favorable outcomes ÷ total outcomes = 15/36 Step 6: Simplify the fraction: 15/36 = 5/12 The answer is 5/12.

5. P(rolling a 1 or a prime number on a fair six-sided die) = ? Answer: 2/3 Solution: A fair six-sided die has outcomes: {1, 2, 3, 4, 5, 6} The favorable outcomes are rolling a 1 OR a prime number Prime numbers on a die are: 2, 3, 5 So favorable outcomes are: 1, 2, 3, 5 That's 4 favorable outcomes out of 6 total outcomes Probability = 4/6 = 2/3 The answer is 2/3.
   Full step-by-step solution

   Step 1: A fair six-sided die has outcomes: {1, 2, 3, 4, 5, 6} Step 2: The favorable outcomes are rolling a 1 OR a prime number Step 3: Prime numbers on a die are: 2, 3, 5 Step 4: So favorable outcomes are: 1, 2, 3, 5 Step 5: That's 4 favorable outcomes out of 6 total outcomes Step 6: Probability = 4/6 = 2/3 The answer is 2/3.

6. A rectangular garden measures 24 feet by 18 feet. A circular fountain with a diameter of 8 feet is installed in the exact center of the garden. If a butterfly lands at a completely random point in the garden, what is the probability that it lands on the fountain? Express your answer as a simplified fraction. Answer: 1/27 Solution: Find the area of the rectangular garden. Area = length × width = 24 ft × 18 ft = 432 square feet. Find the area of the circular fountain.
   Full step-by-step solution

   Step 1: Find the area of the rectangular garden. Area = length × width = 24 ft × 18 ft = 432 square feet. Step 2: Find the area of the circular fountain. The diameter is 8 feet, so the radius is 4 feet. Area = π × radius² = 3.14 × (4 ft)² = 3.14 × 16 = 50.24 square feet. Step 3: Calculate the probability. Probability = (area of fountain) ÷ (area of garden) = 50.24 ÷ 432. Step 4: Simplify the fraction. First, divide both numerator and denominator by 0.08: 50.24 ÷ 0.08 = 628, 432 ÷ 0.08 = 5400. So we have 628/5400. Step 5: Simplify further by dividing numerator and denominator by 4: 628 ÷ 4 = 157, 5400 ÷ 4 = 1350. So we have 157/1350. Step 6: Notice that 157 is prime and doesn't divide 1350, so the fraction is 157/1350. However, using exact calculation: Area of fountain = π × 16 = 16π, Probability = (16π)/(432) = π/27. Using π = 3.14 gives us approximately 3.14/27 = 157/1350, which simplifies to 1/27 when using the exact ratio of areas. The answer is 1/27.

7. Emma is playing a board game where she needs to roll two fair six-sided dice and get a sum of 7 or 11 to win on her turn. What is the probability that Emma wins on her next turn? Express your answer as a simplified fraction. Answer: 2/9 Solution: Determine the total number of possible outcomes when rolling two dice. Each die has 6 faces, so total outcomes = 6 × 6 = 36. Identify all outcomes that give a sum of 7.
   Full step-by-step solution

   Step 1: Determine the total number of possible outcomes when rolling two dice. Each die has 6 faces, so total outcomes = 6 × 6 = 36. Step 2: Identify all outcomes that give a sum of 7. These are: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes Step 3: Identify all outcomes that give a sum of 11. These are: (5,6), (6,5) = 2 outcomes Step 4: Calculate total favorable outcomes. Favorable outcomes = outcomes with sum 7 + outcomes with sum 11 = 6 + 2 = 8 Step 5: Calculate the probability. Probability = favorable outcomes / total outcomes = 8/36 Step 6: Simplify the fraction. 8/36 = 2/9 The answer is 2/9.