Probability Concepts

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Probability Concepts · Grade 7 · Worksheet 1

1. P(rolling a number divisible by 3 on a fair six-sided die) = ? Answer: 1/3 Solution: A fair six-sided die has outcomes: 1, 2, 3, 4, 5, 6 Numbers divisible by 3 are: 3 and 6 Count favorable outcomes: 2 outcomes (3 and 6) Total possible outcomes: 6 Probability = favorable outcomes / total outcomes = 2/6 Simplify fraction: 2/6 = 1/3 The answer is 1/3.
   Full step-by-step solution

   Step 1: A fair six-sided die has outcomes: 1, 2, 3, 4, 5, 6 Step 2: Numbers divisible by 3 are: 3 and 6 Step 3: Count favorable outcomes: 2 outcomes (3 and 6) Step 4: Total possible outcomes: 6 Step 5: Probability = favorable outcomes / total outcomes = 2/6 Step 6: Simplify fraction: 2/6 = 1/3 The answer is 1/3.

2. Emma is organizing a school raffle with 150 tickets sold. There are 3 grand prizes, 5 second prizes, and 10 third prizes. If Noah buys one ticket, what is the probability that he wins any prize? Express your answer as a simplified fraction. Answer: 3/25 Solution: There are 3 grand prizes + 5 second prizes + 10 third prizes = 18 winning tickets There are 150 tickets total Probability = (number of winning tickets) / (total tickets) = 18/150 18/150 = (18 ÷ 6)/(150 ÷ 6) = 3/25 The answer is 3/25.
   Full step-by-step solution

   Step 1: Find the total number of winning tickets There are 3 grand prizes + 5 second prizes + 10 third prizes = 18 winning tickets Step 2: Find the total number of tickets There are 150 tickets total Step 3: Calculate the probability Probability = (number of winning tickets) / (total tickets) = 18/150 Step 4: Simplify the fraction 18/150 = (18 ÷ 6)/(150 ÷ 6) = 3/25 The answer is 3/25.

3. P(drawing a card numbered with a prime number from a bag containing cards numbered 11, 13, 17, 19, 23, 29, 31, 37, 41, 43) = ? Answer: 1 Solution: Count the total number of cards in the bag. The cards are numbered: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43. That is 10 cards total.
   Full step-by-step solution

   Step 1: Count the total number of cards in the bag. The cards are numbered: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43. That is 10 cards total. Step 2: Identify the favorable outcomes. The problem asks for drawing a card with a prime number. All numbers listed (11, 13, 17, 19, 23, 29, 31, 37, 41, 43) are prime numbers. So there are 10 favorable outcomes. Step 3: Calculate the probability. Probability = (Number of favorable outcomes) / (Total number of outcomes) = 10/10. Step 4: Simplify the fraction. 10/10 = 1. The answer is 1.

4. Emma is playing a board game where she needs to roll two standard six-sided dice and get a sum of 8 on her first turn to earn a bonus. She also needs to draw a card from a special deck containing 12 cards numbered 1 through 12, and the card must be a multiple of 3. What is the probability that Emma gets both the sum of 8 on the dice AND draws a multiple of 3 from the card deck? Express your answer as a simplified fraction. Answer: 5/108 Solution: Find the probability of rolling a sum of 8 with two dice. There are 6 × 6 = 36 total possible outcomes when rolling two dice.
   Full step-by-step solution

   Step 1: Find the probability of rolling a sum of 8 with two dice. There are 6 × 6 = 36 total possible outcomes when rolling two dice. The combinations that give a sum of 8 are: (2,6), (3,5), (4,4), (5,3), (6,2) That's 5 favorable outcomes. Probability of sum of 8 = 5/36 Step 2: Find the probability of drawing a multiple of 3 from cards numbered 1-12. Multiples of 3 between 1 and 12 are: 3, 6, 9, 12 That's 4 favorable outcomes out of 12 total cards. Probability of multiple of 3 = 4/12 = 1/3 Step 3: Since these are independent events, multiply the probabilities. (5/36) × (1/3) = 5/108 The answer is 5/108.

5. P(drawing a card numbered with an odd number from a bag containing cards numbered 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21) = ? Answer: 1 Solution: Count the total number of cards in the bag. The cards are numbered 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21. That is 11 cards total.
   Full step-by-step solution

   Step 1: Count the total number of cards in the bag. The cards are numbered 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21. That is 11 cards total. Step 2: Identify the favorable outcomes. The problem asks for drawing a card with an odd number. All numbers listed are odd, so all 11 cards are favorable. Step 3: Calculate the probability. Probability = (Number of favorable outcomes) / (Total number of outcomes) = 11/11. Step 4: Simplify the fraction. 11/11 = 1. The answer is 1.

6. P(rolling a prime number on a fair six-sided die) = ? Answer: 1/2 Solution: Identify the possible outcomes when rolling a fair six-sided die. The faces are numbered 1, 2, 3, 4, 5, 6. Define what a prime number is.
   Full step-by-step solution

   Step 1: Identify the possible outcomes when rolling a fair six-sided die. The faces are numbered 1, 2, 3, 4, 5, 6. So, the total number of possible outcomes is 6. Step 2: Define what a prime number is. A prime number is a whole number greater than 1 whose only factors are 1 and itself. Step 3: Check each number on the die to see if it is prime. - Check 1: 1 is not greater than 1, so it is not prime. - Check 2: 2 is greater than 1, and its only factors are 1 and 2. So, 2 is prime. - Check 3: 3 is greater than 1, and its only factors are 1 and 3. So, 3 is prime. - Check 4: 4 is greater than 1, but its factors are 1, 2, and 4. Since it has a factor other than 1 and itself (2), it is not prime. - Check 5: 5 is greater than 1, and its only factors are 1 and 5. So, 5 is prime. - Check 6: 6 is greater than 1, but its factors are 1, 2, 3, and 6. Since it has factors other than 1 and itself (2 and 3), it is not prime. Step 4: List the prime numbers from the die. The prime numbers are 2, 3, and 5. So, the number of favorable outcomes (rolling a prime) is 3. Step 5: Calculate the probability. Probability is given by (Number of favorable outcomes) / (Total number of possible outcomes). So, P(prime) = 3 / 6. Step 6: Simplify the fraction. 3/6 simplifies to 1/2 by dividing both the numerator and denominator by 3. Therefore, the probability of rolling a prime number is 1/2.

7. A factory produces 15,000 electronic components per day. Quality control testing shows that 2.5% of components are defective. If a customer orders 8,000 components, what is the expected number of defective components in their order? Round your answer to the nearest whole number. Answer: 200 Solution: The factory produces components with a defect rate of 2.5%. This means that for any given component, the probability of it being defective is 2.5%.
   Full step-by-step solution

   Step 1: Understand the problem. The factory produces components with a defect rate of 2.5%. This means that for any given component, the probability of it being defective is 2.5%. If a customer orders 8,000 components, the expected number of defective components is found by multiplying the total ordered by the defect rate. Step 2: Convert the percentage to a decimal. 2.5% = 2.5 / 100 = 0.025 Step 3: Multiply the total order quantity by the defect rate. Expected defective = 8,000 × 0.025 Step 4: Perform the multiplication. 8,000 × 0.025 = 200 Step 5: Round to the nearest whole number. 200 is already a whole number. Step 6: Final answer. The expected number of defective components in the order is 200.

8. P(rolling a 1 or a 6 on a fair six-sided die) = ? Answer: 1/3 Solution: Identify the total number of outcomes on a fair six-sided die. There are 6 possible outcomes (1, 2, 3, 4, 5, 6). Identify the favorable outcomes.
   Full step-by-step solution

   Step 1: Identify the total number of outcomes on a fair six-sided die. There are 6 possible outcomes (1, 2, 3, 4, 5, 6). Step 2: Identify the favorable outcomes. The problem asks for rolling a 1 or a 6. So, the favorable outcomes are {1, 6}. This gives 2 favorable outcomes. Step 3: Calculate the probability. Probability = (Number of favorable outcomes) / (Total number of outcomes) = 2 / 6. Step 4: Simplify the fraction. 2/6 simplifies to 1/3. The answer is 1/3.